Integral Equations and Operator Theory Vol. 13 (1990)

0378-620X/90/040498-5551.50+0.20/0 (c) 1990 Birkhguser Verlag, Basel

B O U N D A R Y I N T E G R A L O P E R A T O R S F O R T H E

HEAT EQUATION

Martin Costabel*

We study the integral operators on the lateral boundary of a space-time cylinder that are

given by the boundary values and the normal derivatives of the single and double layer potentials

defined with the fundamental solution of the heat equation. For Lipschitz cylinders we show

that the 2 x 2 matrix of these operators defines a bounded and positive definite bilinear form on

certain anisotropic Sobolev spaces. By restriction, this implies the positivity of the single layer heat

potential and of the normal derivative of the double layer heat potential. Continuity and bijectivity

of these operators in a certain range of Sobolev spaces are also shown. As an application, we derive

error estimates for various Galerkin methods. An example is the numerical approximation of an

eddy current problem which is an interface problem with the heat equation in one domain and

the Laplace equation in a second domain. Results of numerical computations for this problem are

presented.

1 I n t r o d u c t i o n

Boundary integral equations for linear elliptic partial differential equations have

been studied extensively in theory as well as in practice. For parabolic equations, the study

of several important properties of such boundary integral equations, in particular those

needed for numerical approximations of the solutions, has only recently been started. It

turns out that, for the boundary integral operators of the heat equation, the results are

very similar to the elliptic case. The boundary integral operators have certain anisotropic

ellipticity properties. Their mapping properties are, apart form the complication of the

anisotropy (the equation being of second order in space and of first order in time), in general

even simpler than in the corresponding elliptic case.

Let us look at some of the results that are now well-known for the elliptic case and

compare what is known for the parabolic case and what is shown in this paper.

For smooth domains, the operators in question are pseudodifferential operators.

They are elements of the matrix of operators that constitutes the "Calderdn projector".

The symbol calculus of pseudodifferentiM operators gives an algorithm for determining their

mapping properties modulo smoothing operators. This allows to decide if the operators

"Parts of this work were done while the author had visiting positions at tile Carnegie Mellon University, Pittsburgh, USA, and at the Universi% de Nantes, Prance, or was supported by the DPG-Forschergruppe

KO 634/32-1.

Costabel 499

are Fredholm operators between Sobolev spaces. It also gives a simple criterion for strong

ellipticity. The latter property, namely that the operator can be represented as the sum of

a positive definite and a compact operator, is the basis for the analysis of many numerical

approximation schemes. For elliptic boundary value problems, these tools are now part

of the standard text book literature (see, e. g., [16], [7], [43]). For the parabolic case, an

analogous calculus was constructed by A. Piriou [40], [41]. This fact seems to be widely

unknown, as witnessed by the bibliographies of the papers quoted here. Exceptions are the

books [43], [21]. For the operators we are considering here, Piriou's calculus of parabolic

pseudodifferential operators gives, in the case of a smooth domain, the mapping properties in

the appropriate scale of anisotropic Sobolev spaces (see Proposition 4.3). The explicit forms

of the principal symbols also give a simple explanation for the coercivity (strong "ellipticity"

in an anisotropic sense, here in fact positive definiteness) of the operators of the single layer

potential and of the normal derivative of the double layer potential (see Remark 4.4). When

the coercivity of the single layer potential was found (independently of the author also by

D. N. Arnold and P. Noon [2], [36]), it came as somewhat of a surprise, since it shows that

the degeneracy which is the defining property of parabolic differential operators, disappears

after the boundary reduction.

The important property of coercivity can, in the elliptic case, be proved by two

different methods, namely by explicit computation of principal symbols or by the method of

Green's formula. The first method requires smoothness of the boundary. On the other hand,

it is based on purely mechanical calculations which require no knowledge of the boundary

value problem with which the boundary integral equation is associated. The second method

requires less regularity of the boundary. It is described for general elliptic boundary value

problems in [9], [15] (see also the references given there) and for general transmission prob-

lems in [14], [38].

In the case of second order elliptic systems, the method of Green's formula works

even for general Lipschitz domains (see [10]). It gives an approach to the method of layer

potentials on Lipschitz domains which is different from (and simpler than) the approach by

Meyer, Fabes, Jerison, Kenig, Verchota and others (see [26], [27], [28], [46], [17]). The latter

approach has recently been extended to the case of the heat equation on C 1 cylinders by

Fabes [19], [18] and on Lipschitz cylinders by R. Brown ]5], [6]. In this paper, we present

the variational approach to the layer potentials for the heat equation on Lipschitz cylinders.

From Brown's work [6] we will use his generalization of the Rellich estimates to caloric

functions. These are obtained by elementary applications of Green's formulas and do not

involve any deep harmonic analysis.

Let A be the 2 • 2 matrix of boundary integral operators defined by the heat

single layer potential, the double layer potential, the normal derivative of the single layer

potential, and the normal derivative of the double layer potential on a Lipschitz cylinder

E. We show that A is coercive on the space H-~/2'-1/4(~) • HI/2'~/'(E) (see Theorem 3.11)

which is the natural "energy space" for Cauchy data of the heat equation. By restriction,

one obtains the coercivity for the integral operators of the first kind, defined by the single

layer potential and the normal derivative of the double layer potential, respectively. We also

obtain continuity and bijeetivity for the operator A in a scale of spaces around the energy

space (see Theorem 4.16). As tools and byproducts, we prove a trace lemma for Lipschitz

cylinders (Lemma 4.6), the validity of the classical Green's formulas (Proposition 2.19),

500 Costabel

jump relations (Theorem 3.4) and representations by layer potentials (Theorem 2.20) for the

weak solutions, and solvabihty and regularity results for the variational formulations of the

Dirichlet and Neumann problems with zero initial conditions (Section 4).

These results provide the basis for convergence proofs and error estimates for im-

mericat approximations of the integral equations. Such boundary elemen~ methods for various

initial-boundary value problems for the heat equation have recently received some attention

in the engineering literature (see, e. g., [3], [4], [37]). In Section 5, we describe briefly

boundary element methods for the initial-Dirichlet problem, the initial-Neumann problem,

and mixed problems. Generalizations suggest themselves: Transmission problems, couplings

with finite element methods, and nonlinear boundary value or contact problems of monotone

type have all been studied in the elliptic case with variational arguments. Therefore, for the

heat equation, the analysis of these problems can be based on the results of this paper. We

study in more detail a boundary element method for a transmission problem ("eddy current

problem") where the heat equation in one domain is coupled with Laplace's equation in an-

other domain. This problem has been treated earlier with coupled finite element - boundary

element- ODE methods (see [32], [11]).

For the sake of simplicity we treat only the plain heat equation in this paper. Other

second order parabolic equations can be treated with the same methods as soon as there is

a fundamental solution available. In particular, a first order term with constant coefficients

can be included, and it leads to integral operators which are compact perturbations of those

considered here. Other possible generalizations include the time-dependent Stokes system

(see [25], where the coercivity of the single layer heat potential is generalized to its Stokes

analogue.)

Whereas the variational approach (method of Green's formula) works well for the

integral equations of the first kind, it does not give coercivity results for the classically

preferred integral equations of the second kind. These equations arise if one solves the

Dirichlet problem with a double layer representation or the Neumann problem with a single

layer representation for the solution. If the domain is smooth, one can apply the classical

theory of Fredholm-Volterra integral equations of the second kind with weakly singular kernel

(see [42]). It follows that, for small time intervals, the integral operator defines a contraction,

so that the equation can be solved by a convergent Neumann series. This behavior carries

over to numerical approximations. As one knows from Brown's work [16], the situation

is different as soon as the domain has corners. For a certain class of piecewise smooth

domains, it was shown in [12] that the integral operator N can be decomposed into the sum

of a contraction and a compact operator so that ~I + N is strongly elliptic in a certain

sense and the convergence of numerical approximation methods can be proved. The class

of domains admissible in [12] does, however, not even include all polyhedra in N3. Thus it

remains an open question if, for piecewise smooth or even general Lipschitz cylinders, these

integral operators of the second kind enjoy any coercivity property or if there is another

basis for their use iu boundary element methods.

Another open problem concerns the behavior of the solution near corners and edges.

If the base of the cylinder is a polygon in 2 space dimensions, the asymptotic behavior of the

solution of the initial-Dirichlet problem for the heat equation is known [221, [23]. This can

immediately be translated into the behavior of the solutions of certain boundary integral

equations. Since the behavior is similar to the behavior of solutions of elliptic Dirichlet

Costabel 501

problems at edges in 3 dimensions, it should be obvious how to use this information for

improved boundary element methods using mesh refinements or the singular function method

(For the elliptic case, see [45], [13], [39]). For more general boundary conditions and more

importantly, for 3 space dimensions, this problem remains to be studied.

Contents of this paper:

In Section 2, we collect and improve known facts about Green's formula and the

solvability of the initial-Dirichlet and initiaJ-Neumann problems in variational form on Lip-

schitz cylinders.

In Section 3, we define the layer potentials and the boundary integral operators,

prove the validity of the jump relations in the distr ibutional sense, and we use the results of

Section 2 to prove the coercivity of the boundary integral operators.

In Section 4, we study the regularity of the solutions of the boundary integral

equations in Sobolev spaces. The trace lemma (Lemma 4.6) plays an impor tant role.

In Section 5, we describe the integral equations for Dirichlet, Neumann and mixed

boundary conditions and Galerkin methods with s tandard finite dements for their solu-

tion. V~re give asymptot ic error estimates. A boundary element method for an eddy current

problem is also described here.

In the final Section 6, we give error est imates for this eddy current problem. Then

we describe some details of a numerical implementat ion of this boundary element method.

One impor tant point is the par t ia l integrat ion formula (6.4) for the hypersingular integral.

Finally we present some numerical results for the eddy current problem.

A e k n o w l e d g i n e n t . I want to thank Douglas N. Arnold, Katsuei Onishi and Wolf-

gang L. Wendland for drawing my at tent ion to this subject and for useful discussions.

2 Weak solut ions of the heat equat ion

in Lipschitz cyl inders

Let us first introduce some basic notat ions that will be used throughout this paper.

Let ~t C N* be a bounded domain with Lipschitz boundary F. We assume that

f~ is locally on one side of r and that I" is connected. The dimension n is assumed to

be at least 2, al though the results of this paper, if sui tably interpreted, are also true for

n = 1. The Lipschitz boundary F is locally congruent to the graph of a Lipschitz continuous

function on N,,-1. S tandard references for properties of Lipschitz domains are [33] and [20].

To distinguish the cases of a smooth boundary F and of a Lipschitz boundary I', we write

F ~ C ~ o r F C Lip, respectively.

Unless s ta ted otherwise, we assume F E Lip.

For a number 7' > 0 which will be fixed once and for all, we write

/ : = ( 0 , T) ; Q : = I • E : = I •

Further we define Fit : - {t} x fl, so that

OQ = E U ~o U f~T .

On F there exist the almost everywhere defined outer normal vector field r~ and the surface

measure d(r.

502 Costabel

Standard references for the weak (variational) solution of initial-boundary value

problems for the heat equation are [29], [30, Chap. 3], [31]. There the following anisotropic

Sobolev spaces are introduced:

For r , s >_0let

H'"(~{ z ~{~):= L:(~{; H~(N.')) N H' ( I{ ; L2(t{n))

with the natural norms defined on these spaces of Hilbert space valued distributions. Thus

if

1 /I~ e-it~u(t' x) dt

is the Fourier transform of u with respect to the time variable, we have

lluil~...(,~• = f,~ {li~(-,-)11~,.(.) + (1 + L.I~)'II~(.,-)llb(~)} d..

For r,s <_ 0 we define by duality H~"(PJ ~+1) := (H-~'-'(~'~+I)) '. By H"'(Q) we denote the space of restrictions of elements of H~'S(~ • IR") to Q, equipped

with the obvious quotient norm. The spaces H ' " ( R x F) and H ' " ( E ) are defined analogously.

For I" C C ~ they are defined for any r, s C IR, whereas for r C Lip they are intrinsically

defined for It1 < 1 (s ~ F[ arbitrary), because the spaces H~(F) are invariant against Lipschitz

coordinate transformations if and only if Irl _< 1.

We need the following subspaces:

f t ' " ( Q ) := {u e H~'s((-x~,T) • ft) lu(t,x) = 0 for t < 0} C H~"((-,a:.,T) • a)

and

-~))'"(Q) := { u c H '" ( (0 , oc) • ft) lu(t,x ) --0 for t > T} ,

Ho"(Q ) := L'(I; Ha(a)) N H'(I; L2(a)) c H'"(Q).

- r l , ~

Similarly, H 0 (Q) etc. are defined. We have the duality

/ ; / - " - ' ( Q ) = "(Q) for all r,s with r - 2 r ?7. (2.1)

The spaces / :F"(E) etc. are defined analogously.

Note that in Piriou's papers [4o], [41], the spaces H~'~ are denoted by It ~'2. Another important space is

];(Q) = L'(I; H i ( a ) ) I-] HI(.[; H - I ( a ) )

= {~ ~ L~(I; H~(a)) I 0~ c L~(I; H - ' ( a ) ) }

The subspaces I)(Q) c F((-,:x:,,0) • ft) and F0(Q) c F(Q) are defined analogously. It

follows from the interpolation result [30]

L~(I; X) r3 HI(I; Y) C H](I; [X,Y]}) fl C(I ; [X,Y]}), (2.2)

Costabel 503

where X C Y are Hilbert spaces, that

"I2(Q) is a dense subspace of HI' 89 (2.3)

The role of the Lax-Milgram lemma in the elliptic theory is played in the parabolic

theory by the following generalization ("Lions' projection lemma") whose proof is given in

[29].

L e m m a 2.1 Let H be a Hilbert space and ~ C H a dense subspace. Let a :

H x 9 --~ 1~ be a bilinear form with the properties:

(i) for all ~o E ~, the linear form z ~ a(x, ~) is continuous on H;

(ii) there is a > 0 such that

a(~o, ~) > c~]l~l]~ q for all 9 C ~. (2.4)

Then for each continuous linear form f C H ~, there exists u C H such that

II~ll~ -< ~t/fll-, (2.~)

and

a(u, ~) = <f , ~> for all 9~ C ~. (2.6)

R e m a r k 2.2 The lemma remains obviously true if ~ is not dense in H,

On the Lipschitz domain ft, the classical Green's formulas hold. Thus for u ,v

C2(~), we have

/o Vu(x). vv(x) d~ - f~ o,,u(x)v(x)d~(x) = /o ZXu(x)~(x)dx (2.7)

/o (zX~(x)v(x)- ~(~) A(x)) dx = f(O,~(~)v(x)-~(x)On~(~)) d~(x), (2.S)

where O,u(x) = if(x) 9 Vu(x) is the exterior normal derivative of u on F.

By integrating over (0, T), one obtains from (2.7), if f := (Or - A ) u 6 C(Q):

s w(t , ~). w(t , ~) dx dt + s O,~(t,x)~(t,x)d~dt (2.9)

- f ~ O ~ u ( t , x ) v ( t , x ) d t d ~ = f o f ( t ' z ) v ( t ' x ) d x d t "

Here Vu( t , x) is the gradient with respect to x ~ IR ".

Integration by parts in t from (2.9) gives

/o (VU.VV- UOtv) dxdt § /a uvdz- /aouvdx- s /ofvdxdt. (2.10)

This implies the following weak formulation of the initial-Dirichlet problem with

homogeneous initial and boundary data:

~ ( V u ' V v - uOtv) dxdt + f a T u v d x = ~ f v d x d t (2.11)

for all test functions v E C ~ ( N x gt).

The following lemma is well known. We repeat its proof only to make sure that it.

holds for F C Lip.

504 Costabel

L e m m a 2.3 For all f E H-1'~ = L2(I; H - I ( a ) ) there exists a unique

u C L2(I; H](~)) with Otu E L2(I; H-I(~)) ,

~.e., ~ E Vo(Q), . , , d

Proo f . We take

(2.12)

and

(Or - A ) u = f in Q (2.13)

u = 0 in f't~. (2.14)

1,0 H := H0 (Q) = L2(I; Hl(fl)),

: - {~ : ~ Q I ~ e C ' o ( - ~ , T ) • fl)}

a ( ~ , v ) := fo(Vu.V~-uO,~)dxdt (2.15)

For fixed T C ~, a is continuous in the first variable on H, and we have, in Lemma 2,1.

using ~(T, x) = 0 and Poincar~'s inequality:

fQ 1 s dx T a(9~,~ ) = [Vqol2 d z d t - ~ [TI 2 (2.16) t t=O

1 : f0 Iv l= .dt + s t 12 dx

>- s 2dxdt >

Lemma 2.1 gives the existence of a u ~ H such that

a(u,~o) = <f , ~> for all T E ~. (2.17)

This implies (Or A )u = f in Q. Now from u E H follows Au E L2(I; H- l (a ) ) , hence

0tu = f + A u C L2(I; H-l(f~)).

We see that u E Fo(Q) holds. From (2.2) follows that u i~,,, ~ L2(~2) is well-defined and from

(2.16) that u no = 0. Thus v E ]~0(Q)-

To show uniqueness of the solution, we note that the restrictions of functions in C~(IR; ~)

are dense in Vo, so that the partial integration formula

fQ(Otuv+uOtv) dxdt-- fa uvdz s (2.18)

whose individual terms are all continuous on F0(Q) • Y0(Q), holds by continuity for all

u,v E F0(Q). Therefore our solution u satisfies also (2.11) for all v ~ F0(Q), in particular

f o r "0 ~ l/,:

/Q l f a /Q IV~l~ d~dt + ~ I~1~ d~ = f u d z d t (2.19) T

This implies immediately the uniqueness of u. II

In order to allow inhomogeneous Diriehlet data, we need a trace lemma. We quote

from [31] the following special ease. It is proved in [31] for F C C ~176 but since its statement is

Costabel 505

obviously invariant against Lipschitz coordinate transformations, it is also true for r E Lip

A generalization will be important in Section 4 (Lemma 4.6).

L e m m a 2.4 The trace map 7 : u ~-~ u ~ is continuous and surjective from

/:/ ' ,}(Q) to H 89188

R e m a r k 2.5 Here and in the following the traces on E will always be understood

in the distributional sense, i. e., they are defined by continuous extensions of the trace maps

defined in the pointwise sense for smooth functions. No statements about pointwise a. e.

existence of boundary values of weak solutions will be attempted. This is one of the reasons

why the variational approach is much easier than the LP-harmonic-analysis approach by

Fabes, Brown and others.

Let us also note the following simple observation.

L e m m a 2.6 The bilinear form

:= f O t u ( t , z ) v ( t , z ) d x d t (2.20) d(u, v) d l { x f l

has a continuous extension from C~(l:[ "+a) • C~(~:~ n+l) to

H}(IR; L2(f~)) x H}(IR; L2(f~)), and it holds

d(u ,v) = - d ( v , u ) for al lu , v C H 89 L'(ft)).

We shall use this lemma in particular for u e /:/t '}(Q) and v 6 (//)I, 89 Then

d(u ,v ) = d(uo, vo), where uo,vo C HI'}(IR • f~) are the extensions of u and v by zero for

t < 0 or t > T, respectively.

R e m a r k 2.7 One constant source of difficulties in the variational treatment of the

heat equation is the fact that the term

d(u ,v ) = < % , , v>

does not make sense for arbitrary u, v E ~I ,}(Q) in the case of a finite time interval I =

(0, T). This is shown by functions u, v behaving like log llog(T - t)I near T. Therefore one

has to be very careful when one interprets Green's formulas for weak solutions. Even the

definition of the normal derivative on the boundary for weak solutions is nontrivial.

Now we need the solvability of the Dirichlet problem with homogeneous boundary

data in the space HI' 89

L e m m a 2.8 For all f E /~/-1, 89 thcre exists a unique u C /7/0~'}(Q) such that

(cOt- A ) u = f .

506 Costabel

Proof .

we know that

Ot - A : fdo(Q) -+ L2(I; H-l(f~)) is an isomorphism.

By time-reversal we obtain similarly that

(T) - o ~ - a : V o (Q) - - L~(L H - * ( a ) )

where (T) v 0 (O) = {~ ~ v0((0, ~ ) • a) l v ( t , ~ ) = 0 for t > T } .

Transposition of (2.22) shows that

o ~ - a : L~(I; H ~ ( a ) ) - ~ ( (Q)

Now we interpolate between (2.21) and (2.23).

We use the interpolation result

to obtain

We use the technique of transposition and interpolation as in [31]. From Lemma 2.3

(2.21)

Similarly,

so that indeed

is an isomorphism, (2.22)

is an isomorphism. (2.23)

[L~(:; X), iV(:; V)]} = ~ ( I ; IX, Z]})

, 0o~'}(O) [l)o(Q), L2(I; H~(ft))]~ =

L~(I; H-~(f~)), (~')o (Q) = Ho'~(Q) = H-~'-~(Q), 1

(2.24)

(2,25)

" y u

P r o o f . Let u0 E /7/1, 89 be an extension of 9 to Q which exists according to Lemma 2.4:

.Uo = 9. Then AUo C L2(I; H-I(Q.))

and Otuo C [ / - 89 L2(~2)) = }(I; L2(a)) , (compare Lemma 2.6). This shows that

fo := (0, --A)Uo ~ /7/ a,-}(Q).

_ l

Then Lemma 2.8 gives us a unique ul ~ H~'~(Q) with

(at A)u , = f - fo.

and

= y i,~ Q

9 on E.

is an isomorphism. |

Combining this with the Trace Lemma 2.4, we obtain the solution of the inhomo-

geneous Dirichlet problem.

T h e o r e m 2.9 For every f ff /~/-1,-}(Q) and g C H 89188 there exists a unique

~ ~ ~,~(c2) ~ith

(0 , - a ) u

Costabel 507

Let u := u 0 + u l . Then obviously ( O t - A ) u = f 0 + f - f0 = f and 7u = 7u0 = g. The

uniqueness of u follows from Lemma 2.8. 1

R e m a r k 2.10 The previous lemmas and theorems remain true if the finite interval

I = (0, T) is replaced by ~ + = (0, co) or ~ = ( - o c , oc). In fact, a different possibility for

obtaining them would have been to prove first the simpler results for I replaced by ~ and

then use restriction arguments.

Corollary 2.11 For every f 9 H-I '~ and g 9 H 89188 there exists a unique

u 9 9 ( Q ) ~ i t h (0~ - ~ ) ~ = S a n d ~ u = g

P r o o f . If f is in H-~'~ then the solution u from Theorem 2.8, satisfying Au 9 H-I '~

hence Otu = f + Au 9 H-~'~ belongs to

I?(Q) = {v 9 fI1, 89 lO, v 9 H-~,~

P r o o f .

f in Q; 7u = g which exists according to the previous corollary.

I

C o r o l l a r y 2.12 The trace map 7 : f;(Q) -~ H~' 88 is surjective.

For g 9 H 89188 take the solution u 9 f;(Q) of the Dirichlet problem (0t - A)u =

m

R e m a r k 2.13 It is possible to derive the previous results from Lemma 2.5 to

Corollary 2.12 in a different order. One can first prove the result of Corollary 2.12 for

smooth boundaries, P E C ~176 This can be done by a reduction to the flat case F = Rn-1

and an explicit construction using Fourier transforms, see [36] for details. Corollary 2.12

is invariant against Lipschitz coordinate transformations. Therefore it follows for F E Lip.

Then one can use this to prove Corollary 2.11 directly with the help of Lemma 2.3. Note

that from u0 e F(Q) follows (0t - A ) u 0 9 L2(I; H-I(f~)), so that Lemma 2.3 can be invoked

directly. One does then not need the interpolation arguments (2.24), (2.25).

The surprising result of Corollary 2.12, namely that the dense subspace I)(Q) of

HI' 89 has the same trace space on the boundary as the latter, is one of the key points

for the explanation of the fact that the boundary integral operators for the heat equation,

unlike the heat operator itself, can be bounded and positive in one and the same norm. We

give now an example to show that not all natural generalizations of Corollary 2.10 are true.

Let ft be a la rger domain with ~ C ~ and ~2 := ~ \ ~ . We write accordinglyO

and Q2. The question whether the interior trace mapping

3'~: ] ) ( Q ) ~ H~' 88

is also surjective, can be answered in the affirmative (see also [25]) by choosing an extension

operator R : Hl(f t ) ~ H0~(5) which is also continuous from g - ~ ( f t ) i n t o g - ~ ( 5 ) . For

g 9 H~' 88 then take u 9 ~)(Q) as in Corollary 2.12 and define

a(t, .):= R(u(~, )) for t e ( - ~ , T).

508 Costabel

Then u E /)o(Q) and 7~:~ = g

Another natural construction would be to solve in Qs the Dirichlet problem (at -

A)us = 0 i n Q2,'~'us = g on E andvu2 = 0 on I • 0~. Then us E 1)(Qs), and it seems

natural to conjecture that the definition ~ := u in Q and *k := u2 in Qs would produce a

function t~ E 1)0((~). This is, however, in general not true. Note that the elliptic counterpart

is true, namely

implies ~ E HI(~) .

E x a m p l e 2.14 Let

~,(t, x ) : = e r f (2~ t ) 2

for t > 0, x E ~ . Then u satisfies the initial-boundary value problem

( O t - O ~ ) u ( t , x ) = O f o r x r u(0, x ) - 0 ; u ( t , O ) - 1 .

This is the situation

For a finite T > 0, one has therefore u ~ E H~' 88 -: H 88

Also u E L=((0, T); H~(N)), as is easy to see, e. g., by taking Fourier transforms. One can

also check that

1~1 , ,~1 ~ Otu -- 2,/@t-~e -qw ~_ L2((0, T); H - q ( ~ ) ) (2.26)

and 2

c92u : cgtu ~ t - -~5(x ) C L2((O,T); H-~(~)) (2.27)

hold, where 5(x) is the Dirac delta function. It follows that

c 9(Q) and ~ C 9(O~), but ~ r ~(0) .

The following simple observation will be useful later.

L e m m a 2.15 On the subspace of functions satisfging the homogeneous heat equa-

tion, the norms ofF(Q), of Hl' 89 and of Hl'~ are equivalent. Thus there are constants

rnl, ms, m3 such that

ItuH.~,o(q) _ -~I[~II~,,~(Q) -< msl lul tv(Q) _< m~ll~lI.~,o(q) (2.2S)

for all ~ e F( Q ) ~ t h ( ~ - / ' , ) ~ = 0 in Q.

Costabel 509

P r o o f . The first inequality follows with ml = 1 by the definition of the norms and the

second one with m2 = 1 from the interpolation result (2.2). The nontrivial inequality is the

last one. We use, as we did repeatedly above, that A : H~(it) --+ H-l( f2) is continuous.

From its definition

< A u , ~ > : = - f n V u . V ~ d x f o r ~ e g ~ ( ~ 2 ) ,

we see that it has an operator norm < 1. Hence u e HI'~ implies A u E L2(I; H-1(~2)),

and [[Aul[ H .... (O) ~ I]Ul]gx'~ and with Otu = Au we obtain

2 ---- U 2 0 2 U 2

Iluliv(~) II II.,0(q) + II ,~IIL~(,;. ,(~)) --< 2 II Ll.'0(q)

This is the desired inequality with m3 = k/2. |

In order to study the Neumann problem, we first have to define the normal deriva-

tive on E for weak solutions on Lipschitz domains. In the time independent case, one defines

O,u on F by requiring that Green's formula (2.7) holds for all v C H~(it). This defines

a~u e H-~(F), i f u c H'(~)is given with Au e L2(it) (see [10]). Similarly, we require that

(2.9) should hold. We define

1 1

H '~(Q; c9~ - A) : : {u 9 H~'}(Q) ](0, - A ) u 9 L2(Q)} (2.29)

with the norm Ilull.,,~<~) + 11(4- ~)ulIL~(Q)

We obtain immediately

L e m m a 2.16 The bilinear form

b(u, ~ ) : = f~ ( v u . w - (o, - ~x) u ~) d~d~ + d(u, v) (2.30)

with d(u, v) as defined in Lamina 2.6 is continuous on

H',~(rt • ~; a , - A) • H ' , ~ ( ~ • ~) . For ~,~ 9 C g ( ~ • ~) ~e have

b(u, v) = f~ O,u(t, x)v(t , x) dt &r(z) (2.31)

Def in i t ion 2.17 Let "~- : H 89188 x F) ~ HI ' 89 z it) be a continuous right

inverse of the surjective trace map % For u C HI ' 89 x ~t; 0t - A) we denote by "/~u 9 I 1 1 1

H - ~ ' - z ( E ) the continuous linear form on H~,~(F,), defined by

7,u : ~ ~ b(u ,7-~) . (2.32)

P r o p o s i t i o n 2.18 (i) The map

"~1 : H',}(~t • it; 0, - ~ ) ~ H - ~ ' - ~ ( ~ x r)

is continuous, and, by restriction, the map

1 1 1 1

~1 : H ,~(Q; a, - ~ ) -+ H-~ , -~ (~)

is continuous.

(ii) For u e C2(-O), we have "~,u = O,u .

(2.33)

(2.34)

510 Costabel

Proo f . Clear from Lemma 2.16 and Green's formula (2.9). I

We note now that the first Green formula holds in the form b(u,v) =<71u, 7v>,

or, more explicitly,

[_ w . Wd dt + + [ A)u d d (2.35) 4]R • J]R •

for all u C S 1'} (R • f2; COt - A), v C H 1' 89 ( i t x f~), where d(u, v) is defined by (2.20). The term

d(u,v) on the left hand side of (2.35) does not make sense for arbitrary u ,v E [t1' 89 We

shall see that it does make sense for u E//1, 89 O t - A), v E [t~' 89 and that (2.35) holds

in this case, too. To avoid circular arguments, we present this fact only at the end of this - 1 1

section (Proposition 2.24). By restriction, (2.35) however holds also for u E H ,~(Q; cot - A),

Let v E /2/1,}(~+ x f~) be given and to E I t arbitrary. Define the time-reversal map

nto by

ntov(t, x) := v(to - t, x). (2.36)

(to) 1 1

Then nto v 9 H '~ ( ( -cc , to) x [2), and (2.35) holds. We obtain:

~1 1 P r o p o s i t i o n 2.19 (i) For u 9 H '~(it+ x a; 0l - A ), v 9 I2ii'~(it+ x f~) and any

to 9 I t there holds the first G r e e n f o r m u l a

/o'Os /o Os V u . V(n tov )dxd t + d(u, nto v ) = <71u, Tntov> + ( O t - A ) u n t o v d x d t (2.37)

(ii) I f in addition (cOt - A ) v 9 L2(it+ • f~) holds, then there holds the s e c o n d G r e e n

f o r m u l a

P r o o f . We have to show (ii). This follows from antisymmetrizing (2.37), if we notice that

f ~'ator dx dt = f ~bat,,~ dx dt holds for all ~,, ~b 9 L 2 and that cOt s tov = -- t~tocOtv , s o that

d(u, ,oV) d(v,

holds. I

We are now prepared to generalize the classical representation formula to the case

of weak solutions. Let

G ( t , x ) : = (41rt) ~e ~ 0 ( t ) (2.39)

be the fundamental solution of the heat equation, where ~(t) = 89 +signt) is the Heaviside

f u n c t i o n . Then for u 9 C2([0, oc) x [2) satisfying u(0, x) = 0 on f/0, we have for (t0, x0) C

It+ •

i t ( t 0 , .TO) ~-- / o t ~ G(t 0 -t ,2~ 0 x)(cO,- A ) i t ( t , x )dxd t (2.40)

Costabel 511

Thus u is represented as the sum of a volume potential , a single layer potent ia l and a double

layer potential . We will need this only if (Or - A) u = 0 holds.

If we define the function v by

v( t , x) = G( t , xo - x) , (2.41)

then clearly v 6 / ' /"}(IR+ • ~) and (Of - A ) v = 0 for x r x0. If we therefore define, with

a small e > 0, the domain ~ := gt \ B~(x0), we can use the second Green formula (2.38) on

~ + • ~ . On B~(xo) = {x [ [x - x0[ < r we can use the classical representat ion formula

(2.40). The contributions on ~ + • OB~(xo) cancel, and we obtain the following generalization

of (2.40).

T h e o r e m 2.20 Let u 6 [tl' 89 ( Q ) satisfy (Of- A ) u = 0 in Q. Then there holds the

r e p r e s e n t a t i o n f o r m u l a

u : Ko(~Au)- K,(?u) (2.42)

in Q, where for (to, Xo) 6 Q the single layer potential is defined by

Ko(~)(to, s0) : : <~, ~ ,o~> (2.43)

and the double layer potential by

Kl(w)(to, xo) :- <'7,tCto v, w> (2.44)

with the function v defined in (2.41) which is C ~ in a neighborhood orE.

Returning to the Neumann problem, we find that it is very difficult to prove a

sat isfactory result for weak solutions along the lines of Lemma 2.3 - Theorem 2.9 for the

Dirichlet problem. If one wants to apply Lions' projection lemma (Lemma 2.1), one obtains

the following result whose proof is left to the reader. We will obtain a stronger result later

on by using the method of boundary integral equations (Corollary 3.17).

L e m m a 2.21 For every f ~ L2(Q) and h 6 L2(I; H- 89 there exists a unique

u c ~;(Q) such that

(Ot- A )u = f in Q (2.45)

?lu = h on E. (2.46)

This Neumann problem is given in weak form by

f Q V U . ~ . v d z d t - fQUO~vdxdt= /QfVdxdt § / hvdtd~ (2.47)

for all v C C 3 ~ ( ( - ~ ,, T) • ~ .

The following two lemmas will serve as a certain replacement for the surjectivity

of the map 71. They will be used in the proof of the jump relations of the layer potentials

(Theorem 3.4), compare Lemma 3.3 and Lemma 3.4 in [10].

512 Costabel

L e m m a 2.22 Let C'~(Q) := C~'((O,T] • ~) be the space of restrictwns to Q of

functions in Cg~ x IR'~). Then d ~ ( Q ) is dense in/2/1,~(Q; Ot - A).

P r o o f . The proof follows Grisvard's proof of the elliptic case [20, Lemma t.5.3.9]. The

density of 0~(Q) in / : / 1 ,~ (Q) is clear by tensor product arguments from the density of C ~ ( ~ )

in Hl( f l ) and of 0(7) = C~((0 , T] ) in / : /~ (I) . Similarly, C~((0, T 1 • s is dense in/'/0~' 89 (Q).

Let R be an extension operator from/7/,, 89 into HI' 89 Thus (Ru) Q = u. One can

choose R in such a way that suppRu C [0, cx~) x ~.'~. In this way,/:/a,~(Q) can be identified " 1 L

with a closed subspace of H ,2(~+ • ~'~), and the map u ~ (Ru,(Ot - A)u) identifies -1 1 ~1 1

H '~(Q; 0t - A) with a closed subspace of H '~(~+ • 1~ ~) • L2(l~+ • R'~). Therefore, for - , :

every bounded linear functional l on/'/~' 89 0t - A), there exist f E (H ,2 (~[+ x -- 1 l

H - '-~(~[+ • 1~ ~) and g C L2(~+ • ~'~) such that for all u C ~I, 89 0, ..... A) there holds

<l, u> = <f, Ru> + f g(Ot - A) u dx at. (2.48) J IR

We have supp] C Q and supp 9 C Q. Suppose that <l, ~ > = 0 holds for all ~ 9 C~(Q) . We

have to show 1 = 0.

We have then for all ~ E C~(IR+ • ~ ) :

0 =<l, !o> = <f, ~> + f o g ( O t - A ) ~ d x d t (2.49)

<f, ~p> + fl~, 9(O t - A ) ~ d z d t , : z

where ~ is the extension of 9 by zero outside Q. Now (2.49) means that

f (0t + A)~ (2.50)

1 1 1

holds in ~-+ x ~'~. Since we have f C H - I ' - ~ ( ~ + • ~'~), we find t~ ~ H '~(~+ x ~'~) _ _ (T)

with supp~ C Q, thus g E H ~0'~(Q). It follows that 9 can be approximated by 9~ E 1 1 1

C~~ T) • g/) in the norm of H '~ (Pt+ • f/). It follows (0~ + A) g,~ ~ f in H - t ' - 5 (~.+ •

N~). Hence for any u 9 Ot - A ) there holds

< l , u > = l i 2 ~ { < ( 0 t + A ) g , ~ , Ru> §

-- O.

L e m m a 2.23 The combined trace map ("t',Ta) : u - - (Tu,~/lu) maps ~,o~(~) onto i 1 1 1

a dense subspace of H~, i (~) • H - ~ ' - i ( Z ) ,

P r o o f . This proof follows the proof of its elliptic counterpart in [10, Lemma 3.5]. We

assume that some linear functional (X,@) E H - ~ ' - } ( E ) • H 89 vanishes on the range of

('~,~1), and we have then to show that X and ~b vanish identically.

Costabel 513

Assume therefore

<)~, ~aP> = <r ~/P> for all p C C~(Q) . (2.51)

Let

r = (g ~ Tg) : H~' 88 --~/~I, 89

be the solution operator of the Dirichlet problem

(0 t - - A ) ( T g ) : 0 in Q; ? ( T g ) = 9 on P,,

compare Theorem 2.8. Thus T(~TX) e ///1, 89 Ot - A) with the time-reversal operation ST,

see (2.36). Let

S = ( f ~ S I ) : L2(Q) -~ ft~' 89

be the solution operator of the Dirichlet problem

(0 t - - /~) (S f ) = f in Q; 7(S f ) = 0 on P.,

compare Lemma 2.8.

We can apply the second Green formula (2.38) to u := T(arX) and v := S f with any

f C L2(Q). It follows with to = T

i

= "~"[1 u , " [ t~TV:> - - < ~ U , " [ I ~ T V : >

= - < X , ' ~ S f >

Now, by continuity and Lemma 2.22, (2.51) holds for all p C [-[I' 89 Ot - A), hence for

= S f . Then

<X, ~ l S f > = <g), 3'S f > = 0,

h e n c e fQ u ~ I d~ at = 0 for all f 9 L ~ ( Q ) , h e n c e 0 - u - : r ( ~ ) , h e n c e X = ~ = 0. N o w

(2.51) gives

<r 7 p > - 0 for all p 9189

and because the trace map ~ : /:/~, 89 ---, H 89188 is surjective (see Lemma 2.4), we

conclude that ~ = 0. I

P r o p o s i t i o n 2.24 The first Green formula (2.35) holds for all

u C [/1,~(Q; O t - A ) , v E ~rl, 89 I f a l sov E /:/~' 89 O r - A ) holds, then the Green formula

can also be written in the form

P r o o f . Let u C ~oo(~). Then in the Green formula

valid for v C C ' (Q) , all terms are continuous with respect to v in the /'U' 89 nor,,,.

Therefore (2.53) extends by continuity to v C /:/1, 89 For fixed v e /:/1,~(Q), all terms

in (2.53) except the second one on the left hand side are obviously continuous with respect

514 Costabel

to u in the norm of/:/~' 89 Ot - A). Thus also this term has to be continuous and, by

Lemma 2.22, (2.53) extends to all u ~ ~I~' 89 Ot A). This proves the first assertion.

For u ,v E 0 ~ ( Q ) , (2.52) holds. The term fn~ uv dz is continuous for ~,v in the norm of

/?/~' 89 Ot - A) c )/(Q) c C(I ; L2(a)). This proves the second assertion. I

3 T h e C a l d e r 6 n p r o j e c t o r for t h e h e a t e q u a t i o n

in t h e e n e r g y n o r m

In the preceding section we introduced the single and double layer potentials K0 and

K~ (see (2.43), (2.44)) and showed that every solution u of the homogeneous heat equation

in the space/ t~ '~(Q) can be represented as the sum of a single layer potential with density

in H-~ ' - 88 and a double layer potential with density in H 89188 (see Theorem 2.20).

In this section we study the relations on the boundary that result from the fact that these

densities in the representation formula (2.42) are actually the Neumann and Dirichlet data

of the solution u of the heat equation.

As above, f2 is a Lipschitz domain unless stated otherwise. Most results of this

section are, however, new even for the case of a smooth boundary F.

We begin by rewriting the definition (2.43) of the single layer as follows,

G* ' : (~ v)(t0, ~0)~

Here v is given by (2.41), and 7 '~ is the distribution with support on E, defined by

<7 ~o, X> = <~, 7X> = ~)~ dt do- (3.2)

for all X C C~(N2+1) . The convolution with the fundamental solution G gives the solution

of the initial value problem on N+ x l~ ~,

( O t - A ) u = f in IR+ x lR '~

u(0, x) = 0 for x C ~'~.

Therefore by Fourier transformation in all n + 1 variables, one has

1

5r(G * f ) ( r , ( ) - iv + I~125rf(r 's (3.3)

if .T'u is defined by

7u(~-, ~) = (2~)-~ fr~,,+l e-~("+r x) dt dx.

It follows immediately that the map

(3.4)

(3.5)

Costabel 515

is continuous for any r E tl., where the subscript comp means compact support in the space

variables and loc means the local behavior in the space variables. This is also one of the

simplest examples of an opera tor in the class studied by Piriou [40], [41]. For r = - 1 , the

result follows also from Lemma 2.8.

Now, by duali ty we obta in from Lemma 2.4 for 3" that

I I _ i ~': g -~ ' -~(E) ~ -1,-~ H . . . . p ( I x ~ " ) is continuous. (3.6)

Therefore from the representat ion (3.1) for Ko we obtain:

P r o p o s i t i o n 3.1 The mapping

1 1 _ 1

/ (o : H - g ' - i ( E ) 1,~- (3.7) ---+ Hlo r ( I x ~t")

is continuous.

R e m a r k 3.2 Of course, K0~o satisfies the homogeneous heat equation in I x (~n \

E). By restr ict ion to Q, we have therefore also

i i - 1 !

K o : H - 5 ' - i ( P , ) --+ H ' : (Q; Ot - A) is continuous. (3.8)

In order to get the continuity of the double layer potent ia l operator , we use the

representat ion formula (2.42) for the solution u := Sg of the Dirichlet problem,

( o~ - zx)(Sg) = o in q [ (3.9)

3`(Sg) = g on ~. I

Theorem 2.9 shows that this defines a continuous mapping

i i - i

S : H ~ ' i ( E ) ~ HI'5(Q; Ot A). (3.10)

Now (2.42)gives u = K o ( 7 1 u ) - Kl(TU) or Sg = K o " y l S g - K l g for all 9 E H 89188

hence

Ki = - S + Ko3`1S. (3.11)

Therefore we obta in the mapping properties of K1 by combining those of S (3.10), 3̀ 1 (2.34), and K0 (3.8).

P r o p o s i t i o n 3.3 The mapping

1 1 ~ 1 1

K~ : H~,~(r~) ~ H ,~(Q; 0 , - A) (3.12)

is continuous.

From Proposit ions 3.1 and 3.3 it follows that single and double layer potentials

possess traces 3 ̀ and 0'1 on E. We want to show that for these traces the classical jump

relations are valid. For this purpose we need to consider a complementary domain

f ~ c : = B R \ f 2 and Q C : = I x f ~ C , (3.13)

516 Costabel

where BR := {x C ~ " I lxl < R} is a sufficiently large ball containing P. Then Proposi-

tions 3.1 and 3.3 give also the continuity of the mapping

K0: H- 89 --*//1, 89162 0t - A) (3.14)

and of the mapping 1 1 -1 1 c

K ~ : H ~ , ~ ( X ) ~ H ' ( q ; ~ ~). (3.15)

Therefore both potentials have traces 7 and 71 from both sides on E, and we define the

jumps across E,

[-y~] :== V(,,Q~)--r(~,IQ) (3.16)

[~'1 u] :-- ~'I(U Qc) ~1( u Q)

This does obviously not depend on the choice of BR.

T h e o r e m 3.4 For all ~# ~ H - ~ ' 88 and all w <: H 89 there hold the j u m p

r e l a t i o n s ,

[ 7 K o r 0 ; [,~Kor = --~/,; (3.17)

[ , K ~ ] - ~ ; [ ,~K~w] = 0. (3.1S)

P r o o f . Let %b C H - ~ ' - ~ ( E ) and u = K0•. Then u e /:/-" 89 x BR), hence 7(u [q) = 3'(u Q,:)

by the s tandard trace lemma. From the representation (3.1) of Ko follows that

holds in ~ + x ~:U ~ in the distributional sense. Thus if we choose any test function 9~ C

C ~ ( I x BR), we obtain

<r ~~> - < 7 ' r ~ > = < ( 0 l A ) u , ~ > = - < ~ , (0 , + A ) ~ >

or, with the t ime reversal map KT,

(cgt- A ) T n T u d x d t . (3.19) <2/2, 7 nTgO> . . . . xBR

On the other hand, we can use the second Green formula (2.38) in Q and in Q~ (note that

(0t-A)u=0inQk)Q~):

Q ( o t - - A)99KTudxdt = <71u, 7NT(/O) (-:"~U, ")'IKTW)>

fQ~ (Ot - A) dx dt = <71u, "TnT9 ~> + <"/u, ")'1/r 9~ t Z T ~

Adding both and using [Tu] = 0 - [3'~TW] = [q'~TW], we obtain

f (Or - A) dx dt = <[7,ui, 7~v~> (3.20) ~ T u m

• R

Comparison of (3.19) and (3.20) shows that [71u] = -~P holds.

For the jump relations (3.18) of the double layer potential , we choose w C H~' 88 and

define u := Klw. Then for a test function ~ C C ~ ( ~ + • BR) we have again as above

(Or--- A)cptr = - <[3'1u], 7t~TW> ~, <[Tu], 71t~T~> (3.21)

Costabel 517

Now from the definition of K~ we have K~w = a * (7jw), hence (0t - A)K~w = 7[w in

]R+ x B m hence

fr• (0t - A) dz dt ~o NT u

Comparison of (3.21) and (3.22) gives

= - - flxBR U (6Or + A)t~T~O dx dt

I = < ( 0 t -- A ) U , t~T99> = <"/1 w , N T ~ >

= <W, 71/~T~O> .

(3.22)

<['nu], "/~T~'> = <[-~u] -- w, 71~T~'> (3.23)

for all ~o 6 C~'(~{+ • BR). Now we apply Lemma 2.23 (or rather its time-reversed image)

and conclude that both sides of (3.23) have to vanish identically:

b l u ] - 0 a n d b u ] : w

l

Now we are in the position to define the boundary integral operators.

Def in i t i on 3.5 For r C H-} ' - 88 and w E H}' 88 we define

Vr := 7Kor

N r

1 ( " ~ ( K l W ) Q ~- ~(KI%U) Qc) Kw := ~

W w : : --~fl K 1 w .

R e m a r k 3.6 Due to the jump relations, Theorem 3.4, for V~b and for W w it makes

no differences whether we define the trace from Q or from QC. It is also clear that the choice

of the exterior domain f~c is of no importance for the definitions of the traces.

From the mapping properties of the potentials, Propositions 3.1 and 3.3, together

with the trace lemmas, Lemma 2.4 and Proposition 2.18, we obtain immediately the conti-

nuity of the boundary integral operators in the energy norms.

T h e o r e m 3.7 The operators defined in Definition 3.5 are continuous in the fol-

lowing spaces.

Y :

N :

K :

W :

1 1

/-/-7,- H~'~(2) i(~) --+ ~

1 1

H - ~ ' - H~'~(~;) ~(~) - , ~

HL~(Z) - , H-~,-~(~) 1 1 H-~,-~(E) . H ~ ' ~ ( : ~ ) -~ '

518 Costabel

The jump relations, Theorem 3.4, can be reformulated with the new definitions:

7(//0r 0 = 7(//or = Vr (3.24)

1 1

7, (Kor Q = ~ r 1 6 2 71(Kor Q~ = - ~ r 1 6 2 (3.25)

Q 1 Q': 1 7 ( K , w ) = - w + K w 7(Kl*v) = - w + Kw (3.26)

2 ' 2

71(t(,w)] e = % ( I ( l w ) 0 ~ = - W w . (3.27)

R e m a r k 3.8 These jump relations are well known for smooth domains and smooth

densities. We have proved them here for Lipschitz cylinders and any r C_ H- 89

w E /-/}'{(E). In the smooth case, the operators N and K can, of course, also be defined

intrinsically on E by taking suitable principal values of the singular integrals. For smooth

F, the integrals are actually weakly singular (see, e. g., [42]). For F E Lip and densities

r E L2(E), w E ~l, 89 a corresponding result was shown by Brown [6 I.

We can now take traces in the representation fornmla (2.42) and obtain the relations

that are satisfied on E by the Cauchy data 7u and %u of a solution u of the homogeneous

heat equation. These are the two equations,

1 7u = ~ T U - KTu + V71'u (3.28)

1

71 u = W T u + ~71u + N71u. (3.29)

Thus the Calderdn projector and the associated involution A for the heat equation in Q are

defined as

-1I 1 ( -K I~ ) (3.30) CO := 2 + A := ~ I + W N "

T h e o r e m 3.9 The operator CQ is a projection operator in the space

1, H 89 {(~). (3.31) 'H := H ~ ' i ( E ) •

The following two s tatements for (w, r ) E 7ff are equivalent:

(i) There is a u E fIl ' 89 with (Or - A ) u = 0 in Q and w = 7u, r = 7~u on E.

r = cQ r

Proo f . The implication (i) ~ (ii) is just the derivation of the boundary integral equations

(3.28), (3.29) given above. For the converse implication, we assume w and ~b given and define

u by the representation formula

u := K0r Klw'. (3.32)

Then from the mapping properties of the potentials we know that u C /7/,, 89 holds. As

we have seen, this implies

Costabel 519

The right hand side is equal to (S) according to (ii), which we assumed, hence (i) holds.

The projection property of C O now follows easily: From (3.32) follows (3.33) for any (w, ~).

We just saw that ( ' ~ ) = r (~,~,) holds, hence C~(S) = C O ( : ) f o r any (w, r hence

C~ : Cq. (3.34)

C o r o l l a r y 3.10 The operator .A: ~ ~ 7-l, defined in (3,30), is an isomorphism.

P r o o f . Actually, (3.34) is equivalent to the stronger statement

A -1 = 4A. (3.35)

II

By interchanging the rows of .2,, we define the operator

A : = N W "

Now A is an isomorphism of the space

1 1 1 1

7-/' := H - ~ ' - ~ ( E ) • H~'~(E) (3.37)

onto its dual space 7-/.

In order to state the main result of this section, namely the positive definiteness of

the operator A on 7-{', we define the natural duality between 7-(' and 7-( by

(:) (7 , := <r v> + <W, w> (3.38)

I 1 1 1

for all v ,w c H~,~(r,), ~ , r ~ H-~'-~(Z).

T h e o r e m 3.11 There is a constant a > 0 such that

Ior all

(3.39)

P r o o f . We have to use again a complementary domain f~c and Qc. We could get (3.39)

directly by using the unbounded domain ft c = ]R '~ \ ~. We would then have to define special

Sobolev (Beppo-Levi) spaces on Qc and prove mapping properties of the potential operators

on these dimension-sensitive spaces (see [2] for the case of the operator V in 3 dimensions

and the procedure of [34] for the elliptic case). We prefer here instead, as in (3.13), to use a - - r

bounded ~ := Bn \ f~ and Qn = I • f ~ . We will then first prove a Gs inequality for A

and afterwards derive the positive definiteness from the following lemma which is probably

well known. We include its short proof for convenience.

520 Costabel

L e m m a 3.12 Let A : X -~ X ~ be a bounded linear operator, where X ~ is the dual space of

the Hilbert space X . With a compact operator T : X --~ X ' and a constant c~, let A satisfy

<(A + T)x, x> >_ c~ll~ll~ for all x E X, (3.40)

and

<Ax, x> > 0 for allx ~ X \ {0}. (3.41)

Then there is a constant a~ > 0 such that

<Ax, x> > ct~ ]lxl]~ for all x E X. (3.42)

P r o o f of t h e l e m m a . Assume that (3.42) does not hold. Then there is a sequence

{xk} C X with ]]xklix = 1 and <Ax~,xk:>--~ 0 as k ~ ~c.. We may assume that xk

converges weakly to x E X. Then Txk converges strongly to Tx, and we have

lim <(A + T)xk, xk> = <Tx, x>, k--*oo

in part icular

<Tx, :c> ~ a > O. (3.43)

<(A + T)x, x> - <Tx, x>

lira {<(A + T)x, xk> + <(A + T)x~, x> k ~ o o

- <(A + T ) x , x > - < T x k , xk>}

= lira { - <(A + T ) ( x - xk), ( x - xk)>}

< 0.

From assumption (3.41) follows x = 0 which contradicts (3.43). |

We return to the proof of the theorem.

Let w E H~' 88 and r E H-~ ' - 88 be given. Define u outside of E by the representation

formula

u := K o r Klw. (3.44)

The jump relations, Theorem 3.4, give

[ 'Tu]=-w, [ '71u]=-r (3.45)

From the definition of the boundary integral operators we find

The last two equations allow us to write the bilinear form in (3.39) in terms of the traces of

U :

( ( ; ) A ( : ) ) 1 ({'/lU[Q~ ("/lUlQ~I ' (..f~.~iQ) (~UiQ~ )) (3.47)

On the other hand, we have

<Ax, x> =

Costabel 521

Noting that u satisfies (Or - A ) u = 0 in Q and in Q}I, we can use the first Green formula

(2.35) (see Proposition 2.24),

<~,u o' "~u Q> = f Ivul= dx dt + d(u, u) aO

or, as in (2.52),

<71u Q ' 3'u Q> =

Adding both, we obtain

<3'lu Q , 7u Q>

On Q~, we obtain analogously

/Q lVul= d=dt - d(u,u) + fa~ lul= d=

iQ 1in lu12 dx = IWl ~ d~ dt + 7

>_ s tVul ~dx dt.

(3.48)

1 - is• (3.49)

where O~u is the normal derivative of u on ] x OBR.

Hence (3.47) gives

Now u S• and O,u ISxOBn are defined from (w, %b) by the action of integral operators with

smooth kernels and therefore there is a compact operator T1 : 77 -4 ~ such that

I xOBR U 1

The compact embedding of f/1, 89 L~(Q) together with the mapping properties of/(o and I(I (Propositions 3.1 and 3.3) yield another compact operator T2 : 77 --~ 7-{ such that

uQ~ uQ~

II.,,o(Q) +t lu - < T2 II.,,O(Q~) , >.

Now by Lemma 2.15, for u Q and u Qh the norms in H 1'~ and in H l'} are equivalent, and

therefore we obtain from (3.50)-(3.52)

From the trace lemmas, Lemma 2.4 and Proposition 2.18, we obtain

_< c (11~ IQ IIH,4(Q) + I1~ I<~

522 Costabel

Therefore with ano ther cons tan t c~ > 0 we arrive at

This is the desired G i r d i n g inequality.

In order to show the posi t ivi ty assumpt ion (3.41) of Lemma 3.12, we will show that the last

te rm in (3.50) tends to zero as R ~ oc..

Choose 0 < ao < a such tha t ~ C Bno. Then on QR0 = I x (Bno \ ~), we can use the

second Green formula (2.38) for u and for v as defined in (2.41), where (to, x0) r QR0" It

follows tha t outs ide QRo, in par t icular for ]x] > Ro, the funct ion u coincides with

uo := Kor - KlWo,

where the single and double layer potent ia ls are now defined fron the densities on Eno :--

I x OBno,

W o : = u Y'no , ~ 0 : : C')ru E n ~

The densities wo and r as well as the new boundary En0, are smooth, so that we can now

easily es t imate u ~ = uo ~R and O~u sR = O~uo ~R for R > R0, using the behavior of the

fundamen ta l solut ion G(t, x). From the simple es t imate

IV(t ,z) l _< C, t - " l z l ~"-n for a l l . 9 ~ ,

and a similar one for VG, we ob ta in for finite T,

= CO(R-'~), 0 ,u = O(R- '~-~) , as Izl = R ~ oo,

hence

fz uO, u d t d a = O(R -~ 2) -+ 0 as Izl = R ~ .:x:. x O B n

Thus (3.50) shows tha t l i m n ~ fQ~ IVul 2 dx dt is finite, and tha t

Assume now tha t the right hand side vanishes. Then u(t, .) is constant on f~ and on ~ \

for every t E (0, T). From the heat equa t ion and the ini t ia l condi t ion then follows u - 0 in

I x 1~ ". Hence w = 0 and r = 0. Thus the posit ivi ty assumpt ion of the l em m a is satisfied,

the l emma can be applied, and the proof of the theorem is complete. |

C o r o l l a r y 3 .13 (a) The single layer potential operator

1 1 1 l v : H-~-,-~(r~) ~ H~,~(r)

is an isomorphism, and there is an a > 0 such that

2 for all ~b 9 H - ~ ' - 8 8 < v r r > ~ Hr188 ) (3.56)

Costabel 523

(b) The operator of the normal derivative of the double layer potential

I 1 i 1

w : H~,~(S) ~ H - ~ ' - ~ ( S )

is an isomorphism, and there is an c~ > 0 such that

W 2 1 1 1 <Ww, w> >_ ,~ II I1~,,~(~) for all w E H~'i(E). (3.57)

Proof . The coercivity estimates (3.56) and (3.57) are special cases of (3.30) for w = 0 or

~b = 0, respectively. Together with the continuity of V and W they imply the invertibility

of V and W. 1

Coro l l a ry 3.14 The operators

112 + K ' 2 1 I - 1 ; : H} ' } ( E )~H 89188

~ I + N, - : -~ !1 H-}'- 88 H- 89 N 2

are all isomorphisms.

Proof . The projection property (3.34), or .A 2 = 88 gives the relations

+ : v w , (3.5s)

( ~ I + N ) ( ~ I - N) = WV, (3.59)

from which the corollary follows immediately. II

R e m a r k 3.15 The other two relations in (3.34) can be written as

V - 1 K V = N = W K W -1. (3.60)

Coro l l a ry 3.16 The unique solution u E/]rl, 89 of the Dirichlet problem

( a t - A ) u = oix, Q, I 1

7u = 9 on E with 9 E H~'~(E)

can be represented

(a) as u = K o r KI#, where r E H-} ' -} (E) is the unique solution of the first kind integral

equation

Vr = (~I + K)9. (3.61)

(b) as u = Kor - K19, where r E H-}'- 88 is the unique solution of the second kind

integral equation

( 1 [ _ N)~b = W 9. (3.62)

524 Costabel

(c) as u = K0r where r E H- 89188 is the unique solution of the first kind integral

equation

Vr = 9. (3.63)

(d) as u = K lw , where w E H~' 88 is the unique solution of the second kind integral

equation

( ~ I - K ) w = - g . (3.64)

In (a) and (b), it follows that r = 7au on E.

P r o o f . Obvious from the above uniqueness results and the jump relations in the form

(3.24)-(3.27).

C o r o l l a r y 3.17 The unique solution u 6/:/1, 89 of the Neumann problem

( o , - z X ) u - o i n Q ,

7~u - h o n e w i t h h E H - ~ ' - ~ ( E )

can be represented

(a) as u = K o h - Kaw, where w E H~', ' (E) is the unique solution of the second kind integral

equation

1 (:~I + K ) w = Vh. (3.65)

(b) as u = Koh - K~w, where w E H 89188 is the unique solution of the first kind integral

equation

l I - N) h. (3.66) w ~ (2

(c) as u = KoCh, where ~, E H- 89188 is the unique solution of the second kind integral

equation

( ~ I + N)~b : h. (3.67)

(d) as u = K lw , where w E H 89188 is the unique solution of the first kind integral equation

ww = - h . (3 .68 )

In (a) and (b), it follows that w - 7u on E.

P r o o f . Since a solution u E /2/1, 89 (Q) of (0 t - - ~ k ) U : 0 is in ~'(Q), we can infer its uniqueness

from the variational formulation, Lemma 2.21. The existence of a solution with Neumann

data h E H- 89188 however, could not be obtained from the variational solution, because

the projection lemma, Lemma 2.1, required h E L2(I; H- 89 But now we know that each

one of the four integral equations (3.65)-(3.68) has a unique solution and gives a solution of

the Neumann problem as desired. I

Costabel 525

R e m a r k 3.18 The solution theory of the integral equations in Corollary 3.16 and

3.17 is simpler than that of the corresponding boundary value problems. For the first kind

integral equations (3.61), (3.63), (3.66), (3.68), it is based on positive definiteness, i. e., the

simplest version of the Lax-Milgram lemma. For the Dirichlet problem in Lemma 2.8 and

Theorem 2.9, we had to use Lions' projection lemma and a transposition-and-interpolation

argument to arrive at the "natural" boundary space H} '}(E) . For the Neumann problem,

even this is difficult. For the case of a smooth boundary, the corresponding result on the

Neumann problem can be found in [31], where its (rather sketchy) proof is based on transpo-

sition and interpolation and requires the introduction of another class of (weighted) Sobolev

spaces. For the case of a Lipschitz boundary, that method does not work, because it begins

by considering solutions in H2'I(Q). The classical theory of boundary integral equations for

the heat equation [42] considers the second kind equations (3.62), (3.64), (3.65), (3.67), and

uses the fact that , for the case of smooth F, they are Fredholm-Volterra integral equations

with weakly singular kernels and that therefore their Neumann series are convergent in all

classical function spaces. Brown's work [6] is based on rather elementary applications of

Green's formulas, but then it makes use of the continuity of the operators

N, K : L2(E) --+ L2(E). (3.69)

This continuity property is derived for Y E C 1 by Fabes and Riviere [18] from the Coifman-

McIntosh-Meyer result [8] about the L ~ continuity of the Hilbert transform on Lipschitz

curves. Thus it makes use of deep results of harmonic analysis. The present simple energy

inequality method is better suited as a basis for the analysis of numerical approximations.

4 Regularity

For the case of a smooth boundary F, the theory of the initial-Dirichlet and the

initial-Neumann problem for the heat equation in the whole scale of anisotropic Sobolev

spaces is known [31]. It can also be derived from Piriou's calculus of parabolic pseudodiffe-

rential operators [40], [41], which provides also the corresponding theory for the boundary

integral operators. In this section, we will first quote the relevant theorems for P C C ~.

Then we will show that a part of this regularity theory is still valid for F E Lip. We use two

main tools: An extension of the trace lemma, Lemma 2.5, to more regular Sobolev spaces

(Lemma 4.6) which is analogous to the elliptic case shown in [10, Lemma 3.6], and, secondly,

Rellich estimates that have been proved for the case of the heat equation by R. Brown [61

(see Lemma 4.10).

As above, F is only assumed to be Lipschitz unless explicitly stated otherwise.

Propos i t ion 4.1 [31] Let F E C a

the Dirichlet problem

(0t - A ) u

"~u

i t

Let S be the solution operator 9 H Sg := u of

= 0 i n Q

= g o n e

= 0 on ~o.

526 Costabel

Then for any s >_ O,

is continuous.

P r o p o s i t i o n 4.2 [31] Let F ~ C ~ . Let $1 be the solution operator h ~.~ Slh := u

of the Neumann problem

(St A ) u = 0 i n Q

71u = h on E

u 0 on f~o.

Then for any s > O,

is continuous.

S~: f / - 8 9 [I~+~

P r o p o s i t i o n 4.3 [41] Let F C C ~. Then for any s > O,

v : ~-~+' , ( -~+') / ' ( r~) -~ t?~+' ,c~+')/ ' (s)

J

are isomorphisms.

i thus the tildes can be omitted Note that Ht"(E) = Ht" (E) if and only if )'l < ~, 1 in Proposition 4.3 for Is I < 5.

R e m a r k 4.4 In Piriou's calculus, the principal symbol of the operator V of (an-

isotropic) order - 1 is

~,(V; ~-,~) = (iv + 1~f2) -~

and of the operator W of order +1,

~(W; r ,~) = ( i t + }~12) ~

Thus both symbols are sectorial, i. e., their values are in the sector {z C G II arg z I _< 7r/4}.

Thus their real parts are equivalent to their absolute values:

R e ~ ( V ; r , ( ) _< c l l a ( v ; r , g ) l -< c2(P,l~+l~l) 1 _< c3Rea(V; , - ,~ ) ,

Re~(W; T,() ~ q l ~ ( W ; T,()I < ~2(L~'I~ + I~1) _< c~Re~(W; T,~).

This gives an explanation for tile Gs inequalities satisfied by V and W in the same

spaces where they define continuous bilinear forms (Gs inequality and the whole

concept of strong ellipticity is not mentioned in [41], though). By contrast, for the heat

operator itself, its symbol, iv + ](]2, is not sectorial, and the real part of the symbol, ](12, is

not equivalent to its absolute value, (r 2 + 1(]4) 89

Costabel 527

We return to the case P E Lip. Our trace lamina will follow from an abstract version

which is a generalization of a familiar interpolation lemma (see [30], where the special case

= 0 of the following lemma is shown).

L e m m a 4.5 Let X C Y be Hilbert spaces with X densely embedded in Y .

For a,/3, a,b E FL with a < /3, a < b, consider the space of Y-valued distributions on the

interval (a, b ),

w~,~(~, b) := H~ b); X) n H~((a, b); Y)

Then for 1 1

~ < ~ < / 3 andO := ( ~ - ~ ) / ( / 3 - ~ ) ,

there holds

W~'~(a, b) C C([a, b]; IX, Y]0)- (4.1)

In particular, the map u w+ u(a), defined on C~176 X) , extends to a continuous map

from W='~(a, b) to IX, Y]o.

P r o o f . Using the spectral theorem as in [30], we can assume that Y is a space L2([1, oc); d#)

with a certain positive measure # on [1, oc), and that

x = { f ~ r l A~lf(A)l~dff(A) < ,~}

Then the interpolation spaces are

[X,Y]o = { f E Y I f " A2(1-~ < 00}

for 0 < 0 < 1. We can further assume that (a,b) = ( - , ~ , ~c), and we only need to show

that there is an estimate

o o .

Ilu(0)Hix,Ylo <_ c Ilullw~,~<~> for all u E Co (~t, X). (4.2)

Let 72((, A) be the Fourier transform of u with respect to its first variable. Then u(0, A) =

f ~ ~((, A) d~, and the norms are given by

Ilu(0)ll~x,,-]o -- f l" A20-~176

o o

with

We have

m(,fl ~) := (1 + ,~')~ ~ + (1 + O) ' .

F 1 w i t h C ~ < o c i f a < ~ </3.

The Cauchy-Schwarz inequality gives then

lu(O, A)[ 2 < 0. m(,L A) " o. m(,L A)]'a(,~, )~)12d,~

F < c~,;,o-,~/(~-o~ m(,~,.~)l~(,~, ~)l~d,~. ~ ' - o o

(4.3)

528 Costabel

Mult ip ly ing this inequal i ty with A 2(~-e) = A (2~-i)I(~-~) and in tegra t ing over A, we obta in

(4.2). |

L e m m a 4.6 For Is] < ~, the trace map

7 : 9~+"(~+')/=(Q) -* [r

is continuous.

P r o o f . The s ta tement being local, we can assume tha t the b o u n d a r y P has the form

P = {(z' , z , ) E ~ , - 1 x IR I z , = r with a funct ion ~b which is uni formly Lipschitz

cont inuous on ]R '~-1. We have then

~ ( t , x ' ) ' ' ' = ~ , ( t , x , 0 ) u(t, x , r )) =

with

u,4t , . ) - u(t , x', x~ + ~ ( x ' ) )

Next we in t roduce the no ta t ion for p, q, r E ~ ,

H p'q'r := l iP(I; Sq(~t.; Hr(~Dtn-1))) = Hq(~i{.; }HP(I; Hr(~lff-1))). (4.4)

Thus p denotes the regulari ty in the t ime variable t ff I , q denotes the regulari ty in the

normal variable x~ C ~ , and r denotes the regulari ty in the tangent ia l variables x' C ~ - a

For example, for r > 0 we have then

O~( l ; H ' ( ~ " ) ) = ~ H p''-q'q -- H p?'~ N HP'~

o<_q<_~

Now the map u ~-+ ur which "flattens the boundary" , clearly leaves the spaces H p'q'" invariant

if q,r > 0 and q + r _< 1 hold. On the other hand, it is easy to see (compare [10]) tha t this

map leaves also H p'q'~ invariant for any p and q and tha t it maps H p'qa ;qH p'q+l'~ cont inuously

into H p'qa for any p and q.

We define therefore

{ H p?'~ H p'~ = HP(I ; H~(1R")) for 0 < r ~- I (4.5)

XPX := H p?'~ 71 H px-l ' l for r > 1.

Then the mapp ing u ~ ur is an isomorphism on X p'' and it maps i/p(/; H'(PJ'-~)) con-

t inuously into X p' ' , for any p E ~ , r > 0. Our trace l emma will therefore follow if we show

that the rest r ic t ion to x , = 0 is cont inuous

from X ~ ~ X ~-~'~ to H } + " } + ~ ( I • PQ-~) (4.6)

1 We leave the simpler case - ~ < s < 0 to the reader (this case is also ob ta ined from ).he

smooth case by Lipschitz coordinate t ransformat ions) and assume s > 0. Then (4.5) gives

X ~ N X ~2 ,0 H ~176 if) H ~ if] H ~2 ,0,0 (4.7)

: f t ' ( ~ ; L~(I; H X ( ~ " - ' ) ) ) N H~+'(1R; L ' ( I • 1R" I))

71L2(~; /~/ 89 L 2 ( ~ ' - ~ ) ) r ~ H~+'(IR; L2(I x ~."-~)).

In this way our space is wri t ten as the intersect ion of two W ~'/3 spaces as defined in

Lemma 4.5, namely

Costabel 529

9 W " I + ' ( I { ) with X = L2(I; Ht(n{*- l ) ) ; Y = L2(I x ~1,~-1),

where the restr ict ion to a:~ = 0 is then a continuous map into

[X,Y] 89 s = L2(I; H } + ' ( ~ " - I ) ) , and

9 W~ with X = /7 /}+ , ( i ; L2(]R,~-I)); y = L2(I x ~%,,-a),

where the restr ict ion to xn = 0 is then a continuous map into

'-+'- L=(~{.-I [X,Y]_~, : H, , ( I ; )).

The definition of H}+"}+~(I x R "-1) then shows that the map is continuous as claimed in

(4.6). |

R e m a r k 4 .7 As in the t ime-independent case [10], the endpoint result s = 1 2 remains an open problem.

The generalization to spaces /F ,p (Q) with arbi t rary p is easy:

,y : /7/,.p(Q) _~ /~-},~(~-l)(E ) is continuous (4.8)

1 3 and any E ~ . f o r g < r < g p

T h e o r e m 4.8 For any s 6 (- 89 89 the single layer potential defines a continuous map

K o : H-}+"( -}+s) /2 (E) - -~ /~/1+,,(1+,)/2(Q), (4.9)

and its trace defines a continuous map

V : H -}+" ( -}+ ' ) / 2 (E ) - -+ H}+"( 89 (4.10)

P r o o f . We use the decomposi t ion (3.1),

/4o = GOT ' ,

where 7 ' is the adjoint of the trace map 3'. By Lemma 4.5,

, ' ' ( )' "-1+, , ( -I+,)12 3' : H-~-+"(-~-+')/2(y]) - - + /~l-s,(1-s)/2(Q) C Hcomp ( I X [{n). (4.11)

G is the opera tor of convolution with the fundamenta l solution G, which according to (3.5),

is continuous: - -x+, , ( -1+,) /2

Thus (4.9) holds by composit ion. If we apply Lemma 4.6 once more to V = 3' o K0, we

obtain the continuity of (4.10). |

R e m a r k 4.9 The endpoint result, namely the continuity of

- i 1

v : L2(<, ) -~ H '~(~),

is one of R. Brown's results [6]. He does not prove it, though.

( 4 . 1 2 )

530 Costabel

In order to derive the regularity properties of the other boundary integral operators,

we use a boundary regularity result for the solutions of the initiaLDirichlet and initial-

Neumann problems for the heat equation that was obtained by Brown [61. He generalized

the Rellich(-Payne-Weinberger-Ne~as) estimates that relate the L 2 norms of the tangential

and normal derivatives on the boundary from the elliptic to the parabolic case. Brown shows

that the two norms

tt71UHL~(~ ) and 1[3~ulI~, 89

for solutions of (cqt - A) u = O, u In0 = O. equivalent a r e

We use this result in the following form

L e m m a 4.10 [6] Let S be the solution operator g ~ Sg := u of the Dirichlet

problem

( O t - A ) u = 0 i n Q

7u g on E

u = 0 on f20.

Then

"hS : //~' 89 -* L2(E)

is an isomorphism.

We have shown in Section 3 that 1 I t t

~ s : H~,~(~) ~ H-~ ' -~(~: )

(4.13)

is an isomorphism. We can interpolate between this result and (4.13), but we prefer first to

extend the possible range by transposition.

P r o p o s i t i o n 4.11 Let S be the solution operator of the Dirichlet problem as above.

Then for all s 9 [- 89 89

71S : /'/ 89189 --* /)-~+"(-~+'}/2(E) (4.14)

is an isomorphism.

P r o o f . Let ~T be the time-reversal map

~ u ( t , x) = u (T - t, ~ )

The second Green formula (2.38) shows that the map

~1s : H~,~(~,) --, H-~ ' -~(r~)

is selfadjoint with repect to the duality

(~,,v) ~ <u, ~ v > (4.15)

between H~' 88 and H - ~ ' - [ ( E ) . Therefore, from (4.13) follows by duality:

~ s : L~(~) -* ~ r ( B l ' } ( ~ ) ') = / ~ ~' ~(~) (4 . t6)

is an isomorphism. The result (4.14) then follows by interpolation between (4.13) and (4.16).

|

Costabel 531

R e m a r k 4.12 The tildes in (4.14) are meaningful only for the endpoints s = 4- 89

P r o p o s i t i o n 4.13 For s E ( - 8 9 89 the solution operator of the Dirichlet problem

S : H 89189 ~l+, . ( l+,) /2(Q) (4.17)

is continuous.

P r o o f . We can use a single layer representation for the solution u = Sg of the Dirichlet

problem. According to Corollary 3.16(c), we have for g E H 89188 the relation

Sg = K o V lg.

According to Theorem 4.8, the operator

is c o n t i n u o u s for Isl < ~. "

C o r o l l a r y 4.14 For s E ( - 89 89 the trace map

"~1 : ~ I I + s ' ( l + s ) / 2 ( Q ; Ot - - A ) -~ H- 89189 (4.18)

is continuous.

P r o o f . If we apply the second Green formula (2.38) to v C /:/L 89 Ot - A) and u = Sg 1 1

with g E H~ 'z (E) , we obtain the representation

"y~ - ('y~S)"~ S ' (a , A ) , (4.10)

where " denotes the adjoint with respect to the duality (4.15). The first term on the right

hand side is continuous from /~/l+s,(l+s)/2(Q) to H-~+"(-~+ ' ) /2(E) , according to Lemma 4.6

and Proposition 4.11. The second term maps any/~, ,v(Q; 0t - A) continuously into

H-~+' ,(-~+")/~(E), by Proposition 4.13. I

ous map

Proof .

K1 - S + K071S,

the result follows from Theorem 4.8 and Propositions 4.11 and 4.13.

T h e o r e m 4.16 For any s ff ( 8 9 1 8 9 the operators

C o r o l l a r y 4.15 For any s ff ( 1 1 ~, ~), the double lager potent ial defines a continu-

KI : H~+"(~+' ) /~(E)~/ : / I+, . (~+, ) /~(Q) . (4.20)

Using the representation (3.11),

: H-~+',(-~+') /2(E) _~ H-~+, , ( -~+,) /2(E )

W

2 + K , 2

!i i_i _ N 2 + N , 2

are i somorphisms.

532 Costabel

P r o o f . Immediate from the definitions and the previous propositions. II

R e m a r k 4.17 The continuity of the boundary integral operators in Theorems 4.8

1 is studied by Brown [6] (except for the nonclassical and 4.16 for the endpoint case s =

operator W). It is remarkable that the continuity of the inverses in the endpoint case, that is

the regularity of order 89 of the solutions of the boundary integral equations, is much simpler

to obtain and does not require any deep harmonic analysis, as we shall see.

According to the jump relations, we have

v - ' g = -[~lsg}, (4.21)

where Sg denotes now the solution of a Dirichlet problem in Q and in some exterior cylinder

Q~. For 7xS, we know the endpoint continuity result (4.13). From (3.25), we see that we

can write

(2 I + N) - ' = V - l ( % S ) 1 . (4.22)

Similarly, we have

W - l h = bSlh] , (4.23)

where $1 is the solution operator of the Neumann problem. We have ~/$1 -- (71S) -1, so that

also for 7S1 there holds the endpoint continuity result. Finally there holds

( ~ [ + I 4 ) -1 = - ( 7 S 1 W ) -1 = -W--171S. (4.24)

Combining the relations (4.21)-(4.24), we obtain immediately the following regularity results.

T h e o r e m 4.18 The operators

v_l ~ l ,~ (r ) -~ L2(~)

1_i ~ -1 ' (~ I + K ) < ' (2 - K ) - ~ /t~'}(E) H ' ~ ( E )

+N)-', L2( ) L2(E)

W -1 L2(E) - . H~,~(~)

are continuous.

If the domain ~2 is not just Lipschitz but piecewise smooth, then the above regularity

results can be improved by desribing explicitly the singularities of the solutions of the initial-

boundary value problems at the nonsmooth points of E. For the case when f~ is a polygon in

~2, this has been done by A. Hammoudi [221, [23]. From the form of the first singular function

one sees that the above results, Theorem 4.8, Propositions 4.11 and 4.13, Corollary 4.15,

and Theorem 4.16, remain true in this case, if the interval Is I < 89 is replaced by the larger

interval Is I < So, where So is the smallest of the numbers 7r/w, where w C (0, 2r) runs through

the interior and exterior angles of the polygon P.

If the domain has a "crack" or slit (it is not Lipschitz then), then the endpoint

results s = :k 89 in the above theorems and also Theorem 4.18 cannot be true. The energy

norm results from Section 3 remain valid in this case, however. It can be conjectured that

also all the regularity results for Isl < ~ are still true in this case, but this has not yet been

shown.

Costabel 533

5 Boundary e lement methods

In Section 3, we saw that the initial-Dirichlet problem and the ini t ial-Neumann

problem for the heat equation with no volume sources and zero init ial conditions could be

solved by one of four different boundary integral equations each. The integral equations of

the first kind,

V~ = g (5.1)

for the Dirichlet problem and

W h = ~ (5.2)

for the Neumann problem, are par t icular ly suited for GMerkin approximations. The posi-

t ivi ty of the operators V and W, Corollary 3.13, (3.56) and (3.57), allows very simple error

est imates. The same is true for the integral equation

with the opera tor (matr ix) A defined in (3.36). The operator A contains all four operators

that are needed for the solution of a large class of mixed boundary value and transmission

problems for the heat equation.

For instance, let us consider a mixed Dirichlet-Neumann problem:

Let I ' = F 1 U F 2 , F1AI '2 = 0 , E j = I x r j ( j = 1,2).

Let g E H}'}(E~) and r E H - } ' - } ( E 2 ) be given. We want to solve the problem for u C

/:/',~(Q), (at A ) u = 0 in Q

u -- 0 on f~0 ~u = g on 21 (5.4)

~lu = r on E2.

Let t~ E H 89 be an extension o f g to all of E, and let @ C H - ~ ' - ~ ( E ) be an extension of

r to all of E.

We want to represent u by the representat ion formula (2.42)

u = K o ~ , U - K ~ T u in Q with ~,u = ~ + h, 3,1u = ~ + ~o, (5.5)

where ~o and h are the unknowns, we have to look for

l ! k _ l

H 2" H ~' ~ (5.6)

where the index Ej denotes the subspace of those distr ibutions on E with support in Ej.

Taking traces in (5.5), we obta in the system of boundary integral equations

(,~1 A /h+ - ) ~ + ~ - ) =0. (5.7)

1 1

H - ~'- i and the second If we mult iply the first of these equations with a test function X E ~

one with a test function v E s2 H 2'~ (i. e., we consider the first equation only on E1 and the

second equation only on E2) and observe

<X, h > = 0 = <v, ~>,

534 Costabel

we obtain the weak formulation

,A(:) (:), ; : ' go for all (X,v) C s, x = H ~' ' H ' " (5.8) ' \ r ~2

with

(: :): It is easy to see that (5.8) is an equivalent formulation of the mixed boundary value problem

(5.4). Furthermore, the bilinear form on the left hand side of (5.8) is positive definite on the

space _ k 1 _ t 1

Et X E2 '

since it is the restriction to this subspace of the space 7if' (see (3.37)) of the bilinear form

defined by the operator A which is positive definite according to Theorem 3.11.

Thus all the boundary integral equations (5.1), (5.2), (5.3), (5.8) are defined by pos-

itive definite operators, and therefore every Galerkin method for their solution is convergent.

Let us recall the notion of Galerkin approximation.

Assume that A : X ~ X ' is a linear bounded operator, where X is a Hilbert space

and X ' its dual. Consider the equation

Au = f (5.9)

with its "weak form"

<t, Au> = <t, f > for all t C X. (5.10)

Let XN be a subspace of X. Then XN defines the Galerkin equations for UN ~ XN by

<t, AuN> = <t, f > for all t ~ X N. (5.11)

Now suppose A is positive definite, i. e., there exists a > 0 such that

<A~, v> > ~ Ilvll 2 for all v C X. (5.12)

Then it is a s tandard form of the Lax-Milgram lemma that gives the following result.

L e m m a 5.1 For any f C X' there is a unique solution uN ~ XN of the Galerkin

equations (5.11). There is a constant C, independent o f f and of XN (in fact, C = I+I IAI I /~

will suffice, where ItAII is the operator norm of A), such that

il4 ~NIIx _< c inf{lku vlLx Iv c XN}. (5.13)

R e m a r k 5.2 It is obvious which space plays the role of X in the examples (5.1),

(5.2), (5.3) and (5.8) above.

For the approximating subspaces XN, it is customary to use tensor products of

spaces of functions of the space variables and of spaces of functions of the t ime variable.

Costabel 535

It is then not hard to translate the known approximation properties of the factors into

approximation properties of the tensor product.

Let Xh= C L2(E) and Th, C L2(I ) be families of subspaces for 0 < h , , ht <_ ho, and

let

Vh~,h, := Th, :7: Xh~ (5.14)

be their tensor products. We want to estimate

inf{rlu - vllH,,~(s) l v C Vh~,h,}

for u G H ' " ( E ) with 0 _< r, 0 _< s and p < r, q < s. We give a general approximation result

with a short proof. A more detailed discussion can be found in P. Noon's thesis [36].

Let

PL : L~(I') ~ Xh. and P~ : L2(F) -~ Th,

be the orthogonal L 2 projections. We assume that the following approximation properties

hold.

There is a constant C _> 0 such that for

p := max{Ipl,r}, ~ := max{tql,a} and 0 < ~ < max{0,p}, 0 < A < max{0, q}

II~ - PZ.~'II/~(~) -< C(h~)'-~ll~rlH~(~) for all ~ C H'(F), 0 < h~ _< ho (5.15)

[l~b - P~,r < C(ht)a-)'[I~b[lH,(i) for all r E H~ 0 < ht < ho (5.16)

Such estimates are well known for finite element spaces (spaces of piecewise polynomials)

and for spaces of orthogonal polynomials.

We denote by P~ also the projection defined on L2(E) by

(eLu)(t, x):= (pLy(t, .))(x) for t C Z

and similarly,

( P ~ u ) ( t , z ) := ( P ~ u ( . , z ) ) ( t ) for x E F.

Then P ~ P ~ , = P ~ P ~ is the orthogonal projection in L2(E) onto Vh.,h,.

P r o p o s i t i o n 5.3 Let r > O, s > O, p <_ r, q <_ s, and pq >_ O. Then there is a

constant C > 0 such that

inf{llu - vllH,,~(~) l v ~ V~,h,} ~ C(h~ ~ + hf)FIUIIH--(~)

for all u E H ' " ( E ) , h~, h, E (O, ho). Here

r s

a = m i n { r - p , r - - q ) , / 3 = m i n { s - q , s - - p } . S T

In particular,

i f rq = p s , then a = r - p, /3 = s - q.

(5.17)

(5.18)

(5.19)

536 Costabel

P r o o f . From (5.4) follow easily the ("semi-discrete") approximation properties

Ilu - P~UIiH,,(,~H~(r)) <_ C(h=)P-~tlUHH.(I;Ha(F)), (5.20)

for any # E l~. In part icular , for ~ = I = # = 0, we obtain

5 Chj I I<IL~(r ,H~(r ) ) § Cht'IIP~u[IH,U;L=(I'))

<< C ( h j + ht')llUliH,..,(~).

Now assume first that p, q < 0. Then, as in (5.22), we also have

By duality and the projection property of I p:~ pt - h= h,, we find from (5.22), (5.23),

h P~ ullm,,,,(~ / < C ( h ~ -p ~. hjq)(h~7 + h~')ll~,lbx,.,(,c ). (5.24)

Now from the familiar ("Young's") inequality

a k b k' 1 1 ab .<_ - ~ + V where ~ § ~,7 t, a , b > 0, (5.25)

we find h~-Pht" < C(h~ "-p + ht " -7 ) and ht-qh~" ~ C(h~" =., + ht'-q), so that we obtain

h P~UllH,,,,(z) <_ C(h~ ~ P + h j ~ +ht" q +ht "-~)IluHH,,,(E). (5.26)

This is (5.17), (5 .18) in this case.

Assume now that p, q >_ 0. We use the interpolation result (see also Lemma 4.5),

H'"(E) = C'(I; H ' ( F ) ) C? H'(I; C2(p))

C H" ~ ( [ ; HP(I ' ) )N Hq(1; H~-?(I")).

Then we have from (5.20), (5.21)

- P~I~177 < C(hff -p + h~ ")ll=ll.~,,(=) (5.27)

Iiu - P~ P~ UIIH.,(Z;L2(r)) < C(h~ " -~ -i- h/-q)iiUllH,,,(r,). (5.28)

Adding (5.27) and (5.28), we arrive again at (5.26), hence at (5.17), (5.18). |

with

R e m a r k 5.4 In most cases for the heat equation we wilt have the si tuation (5.18)

p T

q = ~ , s 2

In this case we have

h~ ~ + h, ~ h j -p + h ' ~ ' < C ( h ~ 2 § ht) ~ .

Then for opt imal results we should choose ht ~ h~ 2. In the example in Section 6, we will

encounter also different si tuations, however.

Costabel 537

C o r o l l a r y 5.5 Assume that there are constants cl, c2 > 0 with

clh~ 2 <_ ht <_ c2h~ 2. (5.29)

Then for r > O, p < r, we have under the above assumptions

inf{llu - Vl]g~.~/~(~) l v C Vh~,h~} <<_ C h~ ~-p IlullH.,./~(r~) (5.30)

In the simplest finite element approximations compatible with the energy norm, i. e.,

p = =t=1/2, q = p/2, the space Xh~ consists of piecewise linear functions on a triangulation

of F of meshwidth h~, and Th, of piecewise constant functions on a part i t ion of I = (0, T)

of meshwidth hr. It is s tandard that the hypotheses (5.15), (5.16) then hold for IPl < ~,

Iq] < 89 r < 2, s < 1. If we use this space Vh~,h, for XN in the Galerkin approximations of

the integral equations (5,1), (5.2), or in (5.3) or (5.8) for both components, we can obtain

the following convergence rates (the relation (5.29) between h~ and ht is always assumed):

(a) For smooth F (using the regularity results in Proposition 4.3)

(i) In the norm of the energy space

For(5.1): II~--~N[]H_}_}(~) < Ch~5/~l[gl[~,~(r~) (5.31)

For(5.2): IIh - h~lli, 89 88 <- C h//~l lr (5.32)

For (5.3) and (5.8), there are similar results.

(ii) In negative norms (by application of Aubin-Nitsche duality arguments)

For (5.1): II~ o - ~ll~-~,-=(~) -< Ch211gll~.:(~) (5.33)

For (5.2): Ilh - hNHH .... (X) <_ C h=allr (5.34)

(b) For r C Lip (using the regularity results in Theorem 4.18)

(i) In the energy norms

For (5 . l ) : II~O-~NII H }_ 88 <_ Chxl/2Hgll~,}(x ) (5.35)

For (5.2): II h - hN]IH 89 ) < C h~l/2l[r ). (5.36)

(ii) And the highest orders in negative norms are

For (5.1): II~ - ~N[[~-,,_ 89 <_ Ch~Hg][fzl,~(s) (5.37)

For(5.2): tlh--hNIrL2(~) _< Ch~llr (5.38)

Let us now describe an application which is physically significant and has been

treated with different methods earlier [24], [32], [11]. From the mathemat ica l point of view,

it is interesting because it involves not only all four boundary integral operators from the

Calder6n projector for the heat equation but also the corresponding time-independent opera-

tors for Laplace's equation. It leads therefore to a practical comparison of all these operators.

Another interesting point is the determination of the appropriate space for the weak solution,

538 Costabel

1 o

because it is known from [32] thai. for the normal derivative on E this is H-~ , (E) and not

H - } ' - } ( E ) as for the boundary value problems for the heat equation.

We consider the problem (for n = 2)

(0t - A ) ~ = 0 in Q (5.39)

u = 0 on f~0 (5.40)

Au c = 0 in QC:= I x ( n R J \ ~ ) (5.41)

uC(t,x) : a ( t ) l og lx ] + O(Ix [ 1) as Ixl--+ oc, (5.42)

u u ~ = vo on E (5.43)

O,~u- O,~u c = ~0 on E. (5.44)

In (5.41), the Laplace operator acts on the space variables only.

This interface problem describes an approximation to an eddy current problem.

The domain f~ corresponds to a body with high conductivity surrounded by nonconducting

mater ia l (vacuum) in f~c. Second t ime derivatives in Maxwell 's equations are ignored. The

da ta (v0, ~o0) are the Cauchy da ta of an incident wave generated by some t ime-dependent

external sources that are switched on at t ime t = 0.

We use an integral equation method for the solution of (5.39)-(5.44). The unknowns

a r e

: ~ o n ~-]. " / l u

For u we have in Q the representat ion

and the Calderdn projector on Q gives

(5.45)

In QC we use the representat ion formula for Laplace's equation,

u ~ = - K ~ ( ~ - ~o) + K ~ ( v --vo). (5,48)

Here the harmonic single layer potential K0 a and harmonic double layer potential K1 a are

defined with the fundamental solution

1 log Ixi (5.49)

of the Laplace equation in 1% 2. For x ~ P we have

Ko a := f r e A ( x - z ' )~ (x ' )da (~ : ' ) (5.50)

JI"

u = K o ~ - K lv , (5.46)

the equation (see (3.30))

(;) ( ; - cQ) = 0. (5.47)

Costabel 539

In (5.48), the t ime variable is treated as a parameter.

The Calderdn projector for Laplace's equation on Lipschitz domains has been stud-

ied in [10]. It is defined as

C~=II_A~=II_( -KAvA ) 2 ~ w ~ N ~ , (5.52)

where

(~I- N a ) ~ = -~ l (Ko~) nr

Wav = 71K~v.

If we define again

A ~ := N~ W~ , (5.53)

then A ~ is strongly elliptic but not quite positive definite, since W a maps constant functions

to zero. To avoid unnecessary complications, let us assume that

cap(I ') < 1 (5.54)

holds, where cap(F) denotes the analytic capacity (conformal radius, transfinite diameter).

Under this assumption,

V~ : H - ~ ( F ) ~ H 89 (5.55)

is positive definite and selfadjoint, and

W : H 8 9 H : ~ ( F ) (5.56)

is positive definite modulo constants, i. e., on H~(E)/ IR. The operator A ~ maps H- 89 x

H~( I ' ) continuously onto its dual space, defines a nonnegative bilinear form, and there is an

> 0 such that

( , A a ) _> ~(ll~llH_ 89 IlWliH~(r)/~),

The definition of C~r and the representation (5.48) give the equation

(I-C~)(v~-v~ = 0 ' ~ o (5.58)

where from now on the operators in C~r act on the space variables for each fixed t E I.

Now we subtract (5.47) from (5.58) and arrive at the system

540 Costabel

In (5.57), the duality means integration on I'.

(3.30), we obtain for some a > 0

If we integrate over I and add (5.57) and

1 0 1 1

H-~' (~) c s-s ~, ~(r O.

Now we observe the inclusions

1 1 I 0 H~'~(E) C H~' (E),

T h e o r e m 5.6 The operator

1_ 0 g - ~ ' (E)

A + A A : x

H~,~(r,)

is an isomorphism, and for some a > 0 there holds

1 o H~' (E)

X

H - ~ ~(r~)

Proof .

(5_6o).

(5.60)

(5.61)

(5.62)

The continuity follows from the inclusions (5.61). They also imply (5.63) from

I

i 0 ~ i Theorem 5.7 Let (~o,~o) e U~, (S) • H-~. ~(~) be given. Then the boundary 1_ 0 I 1

integral ~quations (5.59) have a unique solution (~,v) C H-~' (r~) • H~'~(~).

The unique solution u C f I G ( Q ) , u ~ ~ L~(I; H:oJ~)) of th~ interface problem (5.39)-(5.44) has the representation (5.46), (5.48).

Proof . After the preceding discussion, the proof is simple and therefore omitted. It

is simpler than the corresponding proof for general elliptic transmission problems in [14],

because we know already the unique solvability of the integral equations from Theorem 5.6.

II

I 0 I I

T h e o r e m 5.8 Let XN C H-z . (E) • H~'~(E) be any closed subspace. Then there

exists a unique Galerkin approximation ( : ; ) C XN~ solution of

(:) (;:) (:) < ,(A+A ~ ) ( ~ / > : < ,( s+A ~) >forall exN, (5.64) \ V y /

and there is a constant C > O, independent of XN and of (~) C Xsv, such that

11~ -- (r H H - , , o ( E ) 1 - 1 - I I V - - V N I IH2 , , ( E ) ! ! ( 5 . 6 5 )

w ,~ I(r162 _< c inf{ll~, ~IIH- 89 + IIv IL/~=,,(~)

Proof . This is obvious from Theorem 5.6 |

Costabel 541

6 N u m e r i c a l a p p r o x i m a t i o n of an

interface problem

In this section we describe some features of a numerical implementation of the

Galerkin method (5.64) for the boundary integral equations (5.59) for the eddy current

problem (5.39)-(5.44). The matrix of boundary integral operators in (5.59) contains all four

integral operators from the Calder6n projector both for the heat equation and for Laplace's

equation. Therefore any numerical method for (5.59) can, by restriction, also be used to

solve the boundary integral equations for the Dirichlet problem (equations (3.61), (3.62),

(3.63), or (3.64)), for the Neumann problem (equations (3.65), (3.66), (3.67), or (3.68)) and

for mixed boundary value problems (e. g. equation (5.8)) for the heat equation. The same

is true for the corresponding problems of Laplace's equation, and therefore a comparison

of the complexity of the algorithms for the heat equation with those for Laplace's equation

(which have been studied for many years) is also part of the results of this method.

In the implementation of a boundary element method, the most difficult part is

always the computation of the various singular integrals. For the case of Laplace's equation

it is known that the four types of matrix elements appearing in (5.59) can be reduced to the

computation of two types of integrals: On one hand, the operator N a is the adjoint of the

operator K z~, so that

<w, N a ~ > = <~, K a w > , (6.1)

and, on the other hand, the bilinear form

<w, W A y >

with the hypersingular integral operator W zx, can be reduced by partial integration to the

bilinear form with the weakly singular operator V a. Let O~w denote the derivative of w

with respect to the arc length on F (This is for n = 2 space dimensions. The corresponding

results for n = 3 are also well known [33]). Then there holds (see, e. g., [34])

<w, W a y > : <Or VaOTv> for all v , w E H}(F). (6.2)

Thus only the weakly (logarithmically) singular integrals

<~, V ~ >

and the integrals (6.1), which for n = 2 are non-singular for smooth P and strongly singular

at corners of F, have to be computed.

For the case of the heat equation, the operator N of the normal derivative of the

single layer potential is the adjoint of the operator K of the double layer potential with

respect to the "time-twisted" duality (4.15)

~.i'r , N ~ > = <gT~, K w > . (6.3)

There is also an analogue of the partial integration formula (6.2) which reduces the hyper-

singular integrals to weakly singular ones.

542 Cos tabe l

T h e o r e m 6.1 Let O~ and Ot denote the derivative with respect to the arc length on

F and the t ime derivative, respectively. Let ff denote the exterior normal vector. Then

<w, w v > = <a~w, v a r y > - < ~ i , a W ( v l ) > for all ~, ~ r HL~(S) . (6.4)

P r o o f . We show (6.4) for smooth v, w. The first two terms in (6.4) are obviously continuous

on H}' 88 x H} '}(E) . Therefore the last term in (6.4) then has a continuous extension to

this space,too.

Let z , y E P, x # y. Let i ( x ) , i ( y ) and ?(z) , ?(y) denote the normal resp. tangent vactors

at z and at y. Here ? is obtained from ff by a counterclockwise rota t ion by a right angle.

For any 2 • 2 matr ix M with trace t r (M) we have

i ( x ) m M i ( y ) + ~(x)T M y ( x ) -- t r ( M ) ( i ( x ) , g(y)) . (6.5)

Taking for M the second space derivatives of the fundamental solution G(t, x - y), we obtain

= -&(~)O~(y)G(t, x - y) + OtG(t, x - y) ( if(x)- i ( y ) ) (6.6)

Formally, we obta in (6.4) if we mult iply (6.6) by w(s , x ) v ( s - t, y), integrate over E • E and

observe that on F we may always use par t ia l integrat ion for c9~,

<w, 0~r = - <O~w, r for all w, r E H}'~

To make this more precise, we can choose the point x in (6.6) not on F, but on a neighboring

curve F,~ C fl, then integrate over y E F and over x ~ Fro. Here the integrals are all regular.

From the definitions of the traces 7 and 3'1 and the jump relations it follows that if Fm

approaches F in such a way that that the normal vectors converge pointwise, then the normal

derivative of the double layer potential converges in the distr ibutional sense to W v , and the

single layer potential as well as its tangential derivatives converge in the distr ibutional sense

to the respective boundary values (Constructions of suitable approximat ing boundaries for

F e Lip can be found in [35], [20], [46], [6].) For the time derivative cOtG we use the

distr ibutional identi ty

<wl , a~v(~)> : <o, wl, v(~;O>,

where c~tw is the distr ibutional t ime derivative on ~ x F. |

We see that the matr ix elements for the hypersingular operator W require the

computa t ion of two different kinds of weakly singular integrals. If we assume that the

tangential derivative O~ maps the space of tes t-and-tr ia l functions used for the traces of u

on E into the space of those used for the normal derivatives of u on E, then the first term

on the right hand side in (6.4) will be a linear combination of the integrals computed for the

matr ix elements of the weakly singular operator V. So they do not have to be computed by

a separate quadrature. The last term in (6.4), however, requires another quadrature. This

makes the computat ion of the hypersingular integrals the most expensive part of the whole

procedure, even if one takes advantage of the fact that the operator W is selfadjoint. For

a typical run of the program described below, we show in Table 1 the distr ibution of the

computing times for the following five sections of the program:

1. Grid generation, computat ion of all matr ix elements for the four integral operators

from Laplace 's equation, discretization of the given da ta and computa t ion of the right

hand side in (5.59)

Costabel 543

2. Computat ion of the matrix elements for the weakly singular operator V

3. Computat ion of the matrix elements for the operator K

4. Computat ion of the matrix elements for the hypersingular operator W

5. Assembly of the system matrix, solution of the linear system of equations, computation

of errors, output of the results.

Table 1: Relative computing times for 5 program sections, n~ = 12, nt : 16.

Program 1 2 [ 3 4

section

CPU 2.2% 10.6% 40.3% 40.2%

time I

5

6.7%

From these figures it is obvious that in the preparation of further practical appli-

cations of this method, the biggest effort should be put into the improvement of numerical

quadrature for the heat potentials. In comparison, all other considerations, in particular

about clever solution methods for the linear system of equations, are of secondary impor-

tance.

Now we consider the problem (5.59) and its Galerkin approximation (5.64). We

assume that I ~ C N2 is a polygon, and we choose the simplest finite element spaces compatible

with the solution spaces (5.62). Thus we subdivide each of the sides of the polygon r into an

equal number of intervals and denote the total number of intervals by nx, hence h, = Cnx -1.

The time interval [0, T] is subdivided into nt intervals, hence ht = Tnt -1. As described in

Section 5, we choose an approximation for v which is piecewise linear and continuous in space

and piecewise constant in time, and for ~ an approximation which is piecewise constant both

in space and in time. We abbreviate this by writing

Xh. : S ~ x S 1 (I ' ) , Th, : S O [0, T] x S~,[0, T] n ~ n t

x . = (S~ so (r)) • (SO Eo, T], (6z)

The dimension of XN is

N = 2n=nt,

and 1 I I

XN C H-2'~ X H~'v(P~).

We can therefore apply Theorem 5.8 and in particular the error estimate (5.65). We obtain

the existence of a unique Galerkin solution for every choice of n~ and nt. The convergence

rate, however, is not very high for these low order finite element functions. The appearance L 0 I I I

of the norm of H-~ ' (g) instead of H - ~ ' - a ( E ) leads to a loss of $ in the convergence rate:

According to Proposition 5.3, we obtain (for sufficiently smooth io) from (5.17)

inf{H~-r189 l~PE S~ T].'~-~S~ (F)} = O ( h ~ + h t ) , (6.8)

544 Costabel

which is the same order as for the norm in H~176 = L2(E). We do not a t tempt to present

here the regularity theory for the operator A + A ~, so the precise theoretical convergence

order for [[v - VN[[L2(~) is not known. The numerical results (see Figures 2 and 3) are

compatible with a convergence rate of

O(h~ + h,) (6.9)

for both ]]~ - WNI]L2(~) and Ilv - vNllL~(z).

In the following figures, we present numerical results for the case where Y is the

triangle with vertices (0, 0), (1, 0), (0, 2). The time interval is [0, T] = [0, 1]. We choose

(v0, ~o0) as the jumps in the Cauchy data of the function given by

u ( t , ~ ) - 100 _1~-4~1 ~ e for x C ~, the interior of F, (6.10)

4zrt

u( t ,z ) = log]z - - X l ] 2 f o r Z 9 fl*, the exterior of r . (6.11)

Here z0 = (1.5, 2), zl = (0.5, 0.5) (see Figure 1).

2 -

1 -

+

+

1 2

Figure 1: The Example

The exact solution (~, v) is then given by the Cauchy data of the function u:

y = u E ~ So = O n u S .

Note that this example is not particularly pleasant. The functions v and ~ vary rapidly over

[0, T], and the given data (v0, ~o0) do so even more. The code written for these experiments

(partly on workstations of the Andrew system at the Carnegie Mellon University, Pittsburgh,

and partly on a VAX 8530 at the TH Darmstadt,) allows the treatment of any plane poly-

gon and nonuniform meshes. Several experiments were performed for various polygons and

various given data. It was found that the results shown here are typical, in particular with

respect to two observations:

9 Even with a small number N of degrees of freedom, one obtains a reasonable approxi-

mation of the solution over the whole boundary ~, and

Costabel 545

9 The convergence for increasing N is rather slow.

The second point could certainly be improved by choosing higher order elements

or even a variant of an h-p-version of the boundary element method.

In Figures 2 and 3, we plot the relative errors

ll~- v~llv/~./ and I1~ - - ~NllL2(E) IIvlIL'(~ II~ilL~(~

against the number n,2ne which is proport ional to the number of integrals tha t have to be

computed and hence to the computing time. A quasi-uniform mesh on r was used, i. e., a

uniform par t i t ion of the three sides of P. The example problem was solved for

n~ = 6, 9, 12, 18, 24, 30, 36 (numbers on top of the plots)

and for

nt = 4, 8, 16, 32, 64 (numbers on the right hand side of the plots).

The figures show the behavior (6.9) and an order of

2 _ k o((~ ~) ~)

for the most effective choice of the pair (n , , n~). They also show tha t nt should grow faster

than n , , but the relat ion (5.29) of Corollary 5.5 is not necessary.

6"10"'

10''

Q

6 8 12 18 24 30 36 o, , o, ~ , o, o, o, o. 1 0 4

1 82

3 ~ :64

10'' . . . . . . . . i . . . . . . . . i . . . . . . . .

100 1000 10000 100000

nx*nx * nt

Figure 2: Relative L z error for v. Top numbers: n~, right numbers: ne

In Figures 4 and 5, one sees a plot of the exact solution v and ~, respectively, on the

left hand side, and the computed Galerkin solution for the case n~ = 9, nt = 8 on the right

546 Costabel

6"10"

10'-

,r, !

r

, . 3

I l l r

6 9 12 18 24 30 36

1 E l

3

B

16

32

84

1 0 " 2 . . . . . . . . , 9 , . . . . . . . , . . . . . . . .

100 1000 10000 100000

nx*nx " nt

Figure 3: Relative L ~ error for ~. Top numbers: n~, right numbers: nt

hand side. The z axis in the plots describes the boundary F in a non-uniform way: 0 < z < 1

corresponds to the side (0,0)-(1,0), 1 < z < 2 to (1,0)-(0,2), and 2 < x < 3 to (0,2)-(0,0).

We see that even with this small number of mesh points (only 3 per side), one obtains a

reasonable approximation of these rapidly varying functions. Note that the function ~,, being

zero initially, varies from - 7 to +14 after only 3 time steps. The approximation of v is not

so good at the corners of the domain.

From the computed data on the boundary, it is a simple task to compute an ap-

proximation to u or u c for points outside F by using the representation formulas (5.46),

(5.48). This was done for several points in 12 in the above examples. Since one could not,

however, see a conclusive convergence rate, these numbers are not presented here. For the

case shown in Figures 4 and 5, we have for the relative errors

I I ~ ' - ~ l r - ( ~ / I } v l l - r = o . l o l

IF~-~,~IIL~(~)/rI~II~(~) = o.221

l ~ ( T , ~ ) ~ ( T , ~ ) l / l ~ ( T , ~ , ) l = 0.072 for T --- 1, x2 = (0.3,1) (=_ fL

Finally, we want to make some remarks on the computation of the matrix elements

for the Galerkin scheme considered here. One first has to compute the matrix elements for

the four operators A a from potential theory. This is simple, since all these integrals can be

computed analytically. These matrix elements which constitute a (2n, • 2n~) matrix (no

time dependence), are used in two places: First to compute the right hand side of the linear

system of equations from the given data, compare the right hand side of (5.64), and then in

the system matrix in the sum A + A a.

Costabel 547

s.O

S.O

8.0

8,0

. 2 &,O

, ,o

§

I.O

s.O

O

3,0 s

Z.O

Z

Figure 4: v and v N for n~ = 9, nt = 8

The whole system mat r ix is of size

N • N : (2n+n,) • (2n~,) .

But for uniform t ime steps, this mat r ix has a block Toeplitz s tructure due to the convolution

structure of the integral operators. It is also (block) lower t r iangular due to the Volterra

s tructure of the integral operators. Thus only nt blocks of size (2n,) x (2nx) have to be

computed and stored. Only one block of size (2n~) x (2n~) has to be inverted in order to g6t

the solution of the whole system. This shows that even the simplest solution method of order

n , 3 can be used, because the number of integrals that have to be computed is proport ional

to nz2nt, and n t should grow faster than nx as we saw above.

As has been observed [36], [44], in the matr ix elements for A, the t ime integrals

can be computed explicitly. We need the exponential integral function (see [1])

f ~ e tdt

El (z ) = - - - = - E i ( - z ) t

(6.12)

and the following formulas

fo h _ ~ ds 4 ,~ e ,, - (6.13) 9 S 2 d e 4h

f0he ~ ds d) - ~ - - = El( (6.14) s 4h

fo h e - ~ ds = h { e - ~ d d ~ E I ( ~ ) } (6.15)

f o h e _ ~ s d s = ~ { ( 1 d _ ~ d ~ d ~ ) e h + ( ~ ) E I ( ~ ) } . (6.16)

548 Costabel

::~ \ |g,O

'~176 : ! i \ B,O

g

z" o ~176

~2.o i o.O -* . ,0 i

-4,11

-6.O

-%

.o o ' 2. o

1.0

Figure 5: c 2 and ~N for n~ = 9, nt = 8

We give a table of the t ime integrals

k,k' : = ( t -- t ' ) ; TM e - ~ST:N dr' d t J k h . I k ' h

(6.~7)

where ( t - t / ) + = t - t ~ for t > t ' , = 0 for t <_ t ' . One needs these integrals for m = 1 and

rn = 2. It is clear tha t j~,~,h = 0 for k < k', and that j ,~ ,a k ' . ~.~, depends only on I := k -

For k = M , i . e . l = 0 , w e h a v e

d d jl,~k,k h { - e - ~ + (1 + ~ - s } (6.18)

j~,~ 4h d k,~ = ~ - ~ - ~ - E ~ ( 4 h ) (6.19)

F o r k = k ' + l , l > 0, wehave

jl,h a ~ ~_ a ~,~_~ = h { ~ g + l - 1 } ~ ( 4 h ( ~ ) ) , h l -+~+l}E~(4h( t+ l )

4hl

-h(1 - 1)e ,h(~l, _ h(l + 1 ) e - ~ + 2hl~ ,~,

j2,h d d d k,k-I = - E l ( 4 h ( / _ 1)) - E ~ ( ~ ( / + 1) ) + 2Ea(4h~)

4h ,, + d - t ( / - 1)e '"{~" + (l + 1)e ,,,tT+,, _ 2le-~'--~thz}.

In these formulas, for l = 1 the terms containing 1 I are to be omit ted.

) (0.20)

(6.21)

Costabel 549

For the single layer operator V we need the functions jm,h, for m = 1. For the k , k p

double layer K and the normal derivative of the single layer N we need also m = 2. For the

hypersingular operator W, we use the partial integration formula (6,4). The first term on

the right hand side of (6.4) gives rise to linear combinations of integrals that were already

computed for V. In the second term on the right hand side in (6.4), the time derivative is

integrated directly, so that an integral of the form (6.14) remains.

In all these integrals, we have d = Iz - z'] 2, and the remaining integrals over z and

z ' have to take into account the logarithmic singularity of the function E1 at the origin.

We conclude with some observations which are useful for the practical realization:

9 The computing time for the integrals j~,~,h is proportional to the time for evaluations

of the function El. Therefore one should not take a library routine for El, but rather

compute it only to a precision of a few digits. This can speed up the whole program

by several orders of magnitude.

9 On the other hand, formulas (6.20) and (6.21) contain second differences which are

numerically unstable for large l. So for 1 ~ 2 or 3, a suitable quadrature rule should

be applied instead of the exact formulas (6.20), (6.21).

9 The numerical integration of the logarithmically singular functions can be done using a

standard quadrature rule after a suitable transformation of variables. As an example,

an integral of the form

h h

J - - g(l - yt)I( ,y)d dy,

where g has a logarithmic singularity at the origin and f is smooth, can be written as

/07o which can be approximated reasonably well using a standard quadrature rule in the

variables r /and z.

An open problem is the effective numerical approximation of the integrals with the

strong singularities that appear in the matrix elements for the operators K and N at corners

of the domain F.

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Pachbereich Mathematik

Technische Hochschule Darmstadt

SchloflgartenstrM~e 7

D-6100 Darmstadt

W. Germany

Submitted: December 12, 1989