Quadratic C1-spline collocation for reaction-diffusion ... filegeneral theory for spline-collocation...

Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion

Quadratic C1-spline collocation forreaction-diffusion problems

Torsten Linss1 Goran Radojev2 Helena Zarin2

1Fakultät für Mathematik und Informatik, FernUniversität in Hagen, Germany2Department of Mathematics and Informatics, University of Novi Sad, Serbia

"Numerical analysis for Singularly Perturbed Problems"Workshop dedicated to the 60th birthday of Prof. Martin Stynes

TU Dresden, November 16-18, 2011

Outline

Introduction (problem, idea)Layer-adapted meshInterpolation errorCollocation method

StabilityMaximum-norm a priori error boundMaximum-norm a posteriori error boundAn adaptive algorithm

Numerical experimentsConclusion

Reaction-diffusion problem

Reaction-diffusion problemLu := −ε2u′′ + ru = f in (0,1),

u(0) = γ0, u(1) = γ1

ε ∈ (0,1], r ≥ %2 > 0 on [0,1]

0.2 0.4 0.6 0.8 1.0 1.2

¶ = 10-1

¶ = 10-2

Reaction-diffusion problem

Reaction-diffusion problemLu := −ε2u′′ + ru = f in (0,1),

u(0) = γ0, u(1) = γ1

ε ∈ (0,1], r ≥ %2 > 0 on [0,1]

0.2 0.4 0.6 0.8 1.0 1.2

¶ = 10-1

¶ = 10-2

C. de Boor, B. Swartz (SIAM J. Numer. Anal, 1973):general theory for spline-collocation methods applied toclassical (not SP) BVPs

transition to SPBVPs

bounds with “constants” that tend to infinity when ε→ 0

different approach

quadratic C1-splines on a special modified Shishkin mesh

Surla, Uzelac (ZAMM, 1997):quadratic C1-spline collocation, layer-adapted mesh, nodalbasis, mesh points as dof’s⇒ O(N−2 ln2 N)

LRZ (submitted to NA, 2011):B-spline basis⇒ O(N−2 ln2 N)

different approach

Properties of the exact solution

The Green’s function

‖rG(x , ·)‖1 ≤ 1, ‖Gξ(x , ·)‖1 ≤ (%ε)−1 , ‖Gξξ(x , ·)‖1 ≤ 2ε−2

Lemma (Derivative bounds)

Let r , f ∈ C4[0,1]. Then∣∣u(k)(x)∣∣ ≤ C

{1 + ε−k e−%x/ε + ε−k e−%(1−x)/ε

for x ∈ (0,1), k = 0, . . . ,4. Furthermore, the solution can bedecomposed as u = v + w0 + w1. For k = 0, . . . ,4, the regularsolution component v satisfies

∥∥v (k)∥∥∞ ≤ C, while for the layer

parts w0 and w1 we have∣∣w (k)0 (x)

∣∣ ≤ Cε−k e−%x/ε,∣∣w (k)

1 (x)∣∣ ≤ Cε−k e−%(1−x)/ε, x ∈ [0,1].

Properties of the exact solution

The Green’s function

‖rG(x , ·)‖1 ≤ 1, ‖Gξ(x , ·)‖1 ≤ (%ε)−1 , ‖Gξξ(x , ·)‖1 ≤ 2ε−2

Lemma (Derivative bounds)

Let r , f ∈ C4[0,1]. Then∣∣u(k)(x)∣∣ ≤ C

{1 + ε−k e−%x/ε + ε−k e−%(1−x)/ε

for x ∈ (0,1), k = 0, . . . ,4. Furthermore, the solution can bedecomposed as u = v + w0 + w1. For k = 0, . . . ,4, the regularsolution component v satisfies

∥∥v (k)∥∥∞ ≤ C, while for the layer

parts w0 and w1 we have∣∣w (k)0 (x)

∣∣ ≤ Cε−k e−%x/ε,∣∣w (k)

1 (x)∣∣ ≤ Cε−k e−%(1−x)/ε, x ∈ [0,1].

Smoothed Shishkin mesh

Shishkin-mesh transition point

λ := min{σε

%ln N,q

}, q ∈ (0,1/2), σ > 0

The mesh ∆ : x0 < x1 < · · · < xN is generated by xi = ϕ(i/N)with the mesh generating function

ϕ(t) :=

λq t t ∈ [0,q],

κ(t) := p(t − q)3 + λq t t ∈ [q,1/2],

1− ϕ(1− t) t ∈ [1/2,1],

where p is chosen such that ϕ(1/2) = 1/2.Note, that ϕ ∈ C1[0,1] with ‖ϕ′‖∞, ‖ϕ′′‖∞ ≤ C.

Smoothed Shishkin mesh

Shishkin-mesh transition point

λ := min{σε

%ln N,q

}, q ∈ (0,1/2), σ > 0

The mesh ∆ : x0 < x1 < · · · < xN is generated by xi = ϕ(i/N)with the mesh generating function

ϕ(t) :=

λq t t ∈ [0,q],

κ(t) := p(t − q)3 + λq t t ∈ [q,1/2],

1− ϕ(1− t) t ∈ [1/2,1],

where p is chosen such that ϕ(1/2) = 1/2.Note, that ϕ ∈ C1[0,1] with ‖ϕ′‖∞, ‖ϕ′′‖∞ ≤ C.

The interpolation error for piecewise quadratic splines

Notation:

There midpoints of the mesh intervals Ji := [xi−1, xi ] aredenoted with

xi−1/2 := (xi−1 + xi) /2 = xi−1 + hi/2, i = 1, . . . ,N.

For, m, ` ∈ N, m < `, let

Sm` (∆) :=

{s ∈ Cm[0,1] : s|Ji ∈ Π`, for i = 1, . . . ,N

S02 -interpolation

Given an arbitrary function g ∈ C0[0,1], consider theinterpolation problem of finding I0

2g ∈ S02 (∆) with(

i = gi , i = 0, . . . ,N, and(I02g)

i−1/2 = gi−1/2, i = 1, . . . ,N.

Theorem 1

Assume r , f ∈ C4[0,1]. Then the interpolation error u − I02u for

the solution of (1) on a smoothed Shishkin mesh with σ ≥ 3satisfies ∥∥u − I0

2u∥∥∞ ≤ CN−3 ln3 N,

ε2 maxi=1,...,N

∣∣∣(u − I02u)′′

i−1/2

∣∣∣ ≤ CN−2 ln2 N.

S02 -interpolation

2g ∈ S02 (∆) with(

i = gi , i = 0, . . . ,N, and(I02g)

i−1/2 = gi−1/2, i = 1, . . . ,N.

Theorem 1

the solution of (1) on a smoothed Shishkin mesh with σ ≥ 3satisfies ∥∥u − I0

2u∥∥∞ ≤ CN−3 ln3 N,

ε2 maxi=1,...,N

∣∣∣(u − I02u)′′

i−1/2

∣∣∣ ≤ CN−2 ln2 N.

S12 -interpolation

2g ∈ S12 (∆) with(

0 = g0,(I12g)

i−1/2 = gi−1/2, i = 1, . . . ,N,(I12g)

N = gN .

Theorem 2

the solution u of (1) on a smoothed Shishkin mesh with σ ≥ 4satisfies

maxi=0,...,N

∣∣∣(u − I12u)

∣∣∣ ≤ CN−4 ln4 N,∥∥u − I12u∥∥∞ ≤ CN−3 ln3 N,

ε2 maxi=1,...,N

∣∣∣(u − I12u)′′

i−1/2

∣∣∣ ≤ CN−2 ln2 N.

S12 -interpolation

2g ∈ S12 (∆) with(

0 = g0,(I12g)

i−1/2 = gi−1/2, i = 1, . . . ,N,(I12g)

N = gN .

Theorem 2

the solution u of (1) on a smoothed Shishkin mesh with σ ≥ 4satisfies

maxi=0,...,N

∣∣∣(u − I12u)

∣∣∣ ≤ CN−4 ln4 N,∥∥u − I12u∥∥∞ ≤ CN−3 ln3 N,

ε2 maxi=1,...,N

∣∣∣(u − I12u)′′

i−1/2

∣∣∣ ≤ CN−2 ln2 N.

Let ∆ be an arbitrary partition of [0,1]. Our discretisation is:Find u∆ ∈ S1

2 (∆) such that

u∆,0 = γ0,(Lu∆

)i−1/2 = fi−1/2, i = 1, . . . ,N, u∆,N = γ1.

Let {ϕi}N+1i=0 be the B-spline basis in S1

2 (∆). Then we mayrepresent u∆ as

u∆ :=N+1∑k=0

αkϕk ,

where the αk are determined by collocation.

Collocation equation is equivalent to

α0 = γ0, [Lα]i−1/2 = fi−1/2, i = 1, . . . ,N, αN+1 = γ1

with α := (α0, . . . , αN+1)T ∈ RN+2 and

[Lα]i−1/2 := − ε2[

2(αi+1 − αi)

hi(hi + hi+1)− 2(αi − αi−1)

hi(hi−1 + hi)

]+ ri−1/2

i αi+1 +(1− q+

i − q−i)αi + q−i αi−1

q+i :=

4(hi + hi+1)and q−i :=

4(hi + hi−1),

for i = 1, . . . ,N and h0 = hN+1 = 0.

Stability

The operator L is not inverse monotone. Nonetheless, we canestablish its maximum-norm stability.

Theorem 3Assume, there exists a constant κ > 0 such that h1 ≥ κh2,hN ≥ κhN−1 and

hi+1,hi−1}≥ κhi , i = 2, . . . ,N − 1.

Then the operator L is maximum-norm stable with

‖γ‖∞ := maxi=1,...,N

|γi | ≤2(1 + κ)

i=1,...,N

∣∣∣∣∣ [Lγ]i−1/2

ri−1/2

∣∣∣∣∣≤ 2(1 + κ)

κ%2 ‖Lγ‖∞ ,

for all γ ∈ RN+20 :=

{v ∈ RN+2 : v0 = vN+1 = 0

Maximum-norm a priori error bound

Theorem 4Let u be the solution of (1) and u∆ its approximation by thecollocation method on a smoothed Shishkin mesh with σ ≥ 4. Ifassumptions of Theorems 2 and 3 hold true, then

‖u − u∆‖∞ ≤ CN−2 ln2 N.

Proof.

‖u − u∆‖∞ ≤ ‖u − I12u‖∞ + ‖I1

2u − u∆‖∞

≤ CN−3 ln3 N + ‖α− β‖∞ , I12u =

N+1∑k=0

βkϕk

‖α− β‖∞ ≤ C maxi=1,...,N

∣∣∣[L (α− β)]i−1/2

∣∣∣= C max

i=1,...,N

∣∣∣ε2(u − I12u)′′

i−1/2

∣∣∣ ≤ CN−2 ln2 N �

‖u − u∆‖∞ ≤ CN−2 ln2 N.

Proof.

‖u − u∆‖∞ ≤ ‖u − I12u‖∞ + ‖I1

2u − u∆‖∞

≤ CN−3 ln3 N + ‖α− β‖∞ , I12u =

N+1∑k=0

βkϕk

‖α− β‖∞ ≤ C maxi=1,...,N

∣∣∣[L (α− β)]i−1/2

∣∣∣= C max

i=1,...,N

∣∣∣ε2(u − I12u)′′

i−1/2

∣∣∣ ≤ CN−2 ln2 N �

‖u − u∆‖∞ ≤ CN−2 ln2 N.

Proof.

‖u − u∆‖∞ ≤ ‖u − I12u‖∞ + ‖I1

2u − u∆‖∞

≤ CN−3 ln3 N + ‖α− β‖∞ , I12u =

N+1∑k=0

βkϕk

‖α− β‖∞ ≤ C maxi=1,...,N

∣∣∣[L (α− β)]i−1/2

∣∣∣= C max

i=1,...,N

∣∣∣ε2(u − I12u)′′

i−1/2

∣∣∣ ≤ CN−2 ln2 N �

Maximum-norm a posteriori error bounds

Theorem 5Let u be the solution of (1) and u∆ its approximation by thecollocation method on an arbitrary mesh ∆. Then

‖u − u∆‖∞ ≤ η(ru∆ − f ,∆)

with η(q,∆) := ηI(q,∆) + η3(q,∆) + η4(q,∆) and

ηI(q,∆) := ‖(I02q − q)/r‖∞,

η3(q,∆) :=2%2 max

i=1,...,N

{|qi − qi−1/2|, |qi−1/2 − qi−1|

}·min {1,hi%/(4ε)}] ,

η4(q,∆) := maxi=1,...,N

|qi−1 − 2qi−1/2 + qi |4%2 .

Proof. q := ru∆ − f

(u − u∆) (x) =

0G(x , ξ)

(L(u − u∆)

)(ξ) dξ

G(x , ξ)[qi−1/2 − q(ξ)

(I02q − q

)(ξ)G(x , ξ) dξ

G(x , ξ)[qi−1/2 −

(I02q)(ξ)]

(u − u∆) (x) =

0G(x , ξ)

(L(u − u∆)

)(ξ) dξ

G(x , ξ)[qi−1/2 − q(ξ)

(I02q − q

)(ξ)G(x , ξ) dξ

G(x , ξ)[qi−1/2 −

(I02q)(ξ)]

(u − u∆) (x) =

0G(x , ξ)

(L(u − u∆)

)(ξ) dξ

G(x , ξ)[qi−1/2 − q(ξ)

(I02q − q

)(ξ)G(x , ξ) dξ

G(x , ξ)[qi−1/2 −

(I02q)(ξ)]

1) ∣∣∣∣∣∫ 1

(I02q − q

)(ξ)G(x , ξ) dξ

∣∣∣∣∣ ≤ ηI(q,∆)

2)N∑

G(x , ξ)[qi−1/2 −

(I02q)(ξ)]

dξ =N∑

whereIi =

G(x , ξ)(ξ − xi−1/2)Ri(ξ) dξ

Ri(ξ) =qi − qi−1

hi+ 2(ξ − xi−1/2

)qi−1 − 2qi−1/2 + qi

1) ∣∣∣∣∣∫ 1

(I02q − q

)(ξ)G(x , ξ) dξ

∣∣∣∣∣ ≤ ηI(q,∆)

2)N∑

G(x , ξ)[qi−1/2 −

(I02q)(ξ)]

dξ =N∑

whereIi =

G(x , ξ)(ξ − xi−1/2)Ri(ξ) dξ

Ri(ξ) =qi − qi−1

hi+ 2(ξ − xi−1/2

)qi−1 − 2qi−1/2 + qi

|Ii | ≤hi

2‖Ri‖∞,Ji

G(x , ξ) dξ

|Ii | ≤hi

{‖Ri‖∞,Ji

|Gξ(x , ξ)| dξ +∥∥R′i

∥∥∞,Ji

G(x , ξ) dξ}

2a,b) ∣∣∣∣∣N∑

∣∣∣∣∣ ≤ 2%2 max

i=1,...,N

2‖Ri‖∞,Ji

+1%2 max

i=1,...,N

8∥∥R′i

∥∥∞,Ji

= η3(q,∆) + η4(q,∆) �

|Ii | ≤hi

2‖Ri‖∞,Ji

G(x , ξ) dξ

|Ii | ≤hi

{‖Ri‖∞,Ji

|Gξ(x , ξ)| dξ +∥∥R′i

∥∥∞,Ji

G(x , ξ) dξ}

2a,b) ∣∣∣∣∣N∑

∣∣∣∣∣ ≤ 2%2 max

i=1,...,N

2‖Ri‖∞,Ji

+1%2 max

i=1,...,N

8∥∥R′i

∥∥∞,Ji

= η3(q,∆) + η4(q,∆) �

|Ii | ≤hi

2‖Ri‖∞,Ji

G(x , ξ) dξ

|Ii | ≤hi

{‖Ri‖∞,Ji

|Gξ(x , ξ)| dξ +∥∥R′i

∥∥∞,Ji

G(x , ξ) dξ}

2a,b) ∣∣∣∣∣N∑

∣∣∣∣∣ ≤ 2%2 max

i=1,...,N

2‖Ri‖∞,Ji

+1%2 max

i=1,...,N

8∥∥R′i

∥∥∞,Ji

= η3(q,∆) + η4(q,∆) �

An adaptive algorithm

Idea: to adaptively design a mesh for which the localcontributions to the a posteriori error estimator

µi (u∆,∆) :=

∥∥∥∥∥ I02q − q

∥∥∥∥∥∞,Ji

+|qi−1 − 2qi−1/2 + qi |

{|qi − qi−1/2|, |qi−1/2 − qi−1|

}]are the same on each mesh interval, i.e.,

µi−1 (u∆,∆) = µi (u∆,∆) , i = 1, . . . ,N.

This is equivalent to

Qi (u∆,∆) =1N

N∑j=1

Qj (u∆,∆) , Qi (u∆,∆) := µi (u∆,∆)1/2 .

Algorithm modification(Kopteva, Stynes SIAM J. Numer. Anal., 2001):

We stop the algorithm when

hiQ̃i (u∆,∆) ≤ γ

N∑j=1

hjQ̃j (u∆,∆) ,

for some user chosen constant γ > 1, where

Q̃i (u∆,∆) :=(

h2i + µi (u∆,∆)

Algorithm (de Boor, 1973)

1. Fix N, r and a constant γ > 1. The initial mesh ∆[0] isuniform with mesh size 1/N.

2. For k = 0,1, . . . , given the mesh ∆[k ], compute thediscrete solution u[k ]

∆[k ] on this mesh using the S12 -collocation

method. Set h[k ]i = x [k ]

i − x [k ]i−1 for each i . Compute the

piecewise-constant monitor function M [k ] defined by

M [k ](x) := Q̃[k ]i := Q̃i

(u[k ]

∆[k ] ,∆[k ])

for x ∈(xk

i−1, xki).

The total integral of the monitor function is

I[k ] :=

0M [k ](t) dt =

N∑j=1

h[k ]j Q̃[k ]

3. Test mesh: If

h[k ]j Q̃[k ]

j ≤ γI[k ]N−1 for all j = 1, . . . ,N

then go to Step 5. Otherwise, continue to Step 4.4. Generate a new mesh by equidistributing the monitor

function M [k ], i.e., choose the new mesh ∆[k+1] such that∫ x [k+1]i

x [k+1]i−1

M [k ](t) dt =I[k ]

N, i = 1, . . . ,N.

Return to Step 2.

5. Set ∆∗ = ∆[k ] and u∗∆∗ = u[k ]

∆[k ] then stop.

Test problem

−ε2u′′(x) + 4u(x) = cos 12x , x ∈ (0,1), u(0) = u(1) = 0

Discrete maximum-norm

‖uε − uε∆‖∞ ≈ χεN := max

i=1,...,Nm=0,...,M

∣∣∣(uε − uε∆) (xi−1 + mM−1hi)∣∣∣

χN := maxk=0,...,20

χ10−k

Rates of convergence

p̃N :=lnχN − lnχ2N

ln 2(Bakhvalov and uniform mesh)

pN :=lnχN − lnχ2N

ln 2 + ln ln N − ln ln 2N(two meshes of Shishkin type)

Test problem

−ε2u′′(x) + 4u(x) = cos 12x , x ∈ (0,1), u(0) = u(1) = 0

i=1,...,Nm=0,...,M

∣∣∣(uε − uε∆) (xi−1 + mM−1hi)∣∣∣

χN := maxk=0,...,20

χ10−k

Test problem

−ε2u′′(x) + 4u(x) = cos 12x , x ∈ (0,1), u(0) = u(1) = 0

i=1,...,Nm=0,...,M

∣∣∣(uε − uε∆) (xi−1 + mM−1hi)∣∣∣

χN := maxk=0,...,20

χ10−k

Table: Maximum-norm errors of the collocation method onlayer-adapted and uniform mesh

smoothed standard Bakhvalov uniformShishkin mesh Shishkin mesh mesh mesh

N χN pN χN pN χN p̃N χN p̃N

26 3.198e-03 2.49 2.879e-03 2.29 1.024e-04 2.07 1.574e-01 0.0027 8.375e-04 2.10 8.375e-04 2.10 2.439e-05 2.03 1.575e-01 0.0028 2.588e-04 2.08 2.588e-04 2.08 5.987e-06 2.01 1.575e-01 0.0029 7.800e-05 2.05 7.800e-05 2.05 1.485e-06 2.01 1.574e-01 0.00210 2.335e-05 2.03 2.335e-05 2.03 3.698e-07 2.00 1.574e-01 0.00211 6.940e-06 2.02 6.940e-06 2.02 9.229e-08 2.00 1.574e-01 0.00212 2.046e-06 2.01 2.046e-06 2.01 2.305e-08 2.00 1.574e-01 0.00213 5.971e-07 2.00 5.971e-07 2.00 5.761e-09 2.00 1.574e-01 0.00214 1.726e-07 2.00 1.726e-07 2.00 1.440e-09 2.00 1.575e-01 0.00215 4.947e-08 2.00 4.947e-08 2.00 3.599e-10 2.00 1.575e-01 0.00216 1.406e-08 2.00 1.406e-08 2.00 8.998e-11 2.00 1.574e-01 0.00217 3.966e-09 2.00 3.966e-09 2.00 2.249e-11 2.00 1.574e-01 0.00218 1.111e-09 — 1.111e-09 — 5.623e-12 — 1.574e-01 —

Table: A posteriori-error estimates for smoothed Shishkin meshes;ε = 10−12

N χN η ηI η3 η4 χN/η

26 3.198e-03 6.468e-02 1.662e-03 5.302e-02 1.000e-02 4.945e-0227 8.375e-04 2.437e-02 2.190e-04 1.991e-02 4.238e-03 3.437e-0228 2.588e-04 8.510e-03 2.730e-05 6.905e-03 1.578e-03 3.042e-0229 7.800e-05 2.807e-03 3.401e-06 2.265e-03 5.386e-04 2.779e-02210 2.335e-05 8.878e-04 4.246e-07 7.138e-04 1.736e-04 2.630e-02211 6.940e-06 2.724e-04 5.302e-08 2.185e-04 5.381e-05 2.548e-02212 2.046e-06 8.168e-05 6.624e-09 6.545e-05 1.623e-05 2.504e-02213 5.971e-07 2.407e-05 8.278e-10 1.927e-05 4.797e-06 2.481e-02214 1.726e-07 6.996e-06 1.035e-10 5.600e-06 1.397e-06 2.468e-02215 4.947e-08 2.011e-06 1.293e-11 1.609e-06 4.017e-07 2.461e-02216 1.406e-08 5.723e-07 1.616e-12 4.579e-07 1.144e-07 2.457e-02217 3.966e-09 1.616e-07 2.021e-13 1.293e-07 3.231e-08 2.455e-02218 1.111e-09 4.529e-08 2.526e-14 3.624e-08 9.058e-09 2.454e-02

Table: A posteriori-error estimates for smoothed Shishkin meshes,robustness of the estimator; N = 214

ε χN η ηI η3 η4 χN/η

1 1.715e-09 8.335e-08 7.159e-13 6.846e-08 1.489e-08 2.058e-0210−2 1.720e-07 6.971e-06 3.587e-12 5.580e-06 1.392e-06 2.468e-0210−3 1.726e-07 6.996e-06 8.304e-11 5.599e-06 1.397e-06 2.468e-0210−4 1.726e-07 6.996e-06 1.013e-10 5.600e-06 1.397e-06 2.468e-0210−6 1.726e-07 6.996e-06 1.034e-10 5.600e-06 1.397e-06 2.468e-0210−8 1.726e-07 6.996e-06 1.035e-10 5.600e-06 1.397e-06 2.468e-0210−12 1.726e-07 6.996e-06 1.035e-10 5.600e-06 1.397e-06 2.468e-0210−16 1.726e-07 6.996e-06 1.035e-10 5.600e-06 1.397e-06 2.468e-0210−20 1.726e-07 6.996e-06 1.035e-10 5.600e-06 1.397e-06 2.468e-02

Table: The adaptive algorithm, ε = 10−12

N χN p̃N η ηI η3 η4 χN/η #iter26 2.24e-04 0.63 2.07e-03 1.07e-04 1.67e-03 2.97e-04 1.08e-01 827 1.45e-04 2.09 1.12e-03 1.43e-05 9.88e-04 1.13e-04 1.30e-01 1428 3.39e-05 2.17 1.26e-04 1.65e-06 1.09e-04 1.46e-05 2.70e-01 629 7.53e-06 2.37 3.96e-05 2.15e-07 3.56e-05 3.77e-06 1.90e-01 5210 1.46e-06 0.80 8.96e-06 2.60e-08 8.07e-06 8.67e-07 1.63e-01 5211 8.37e-07 3.86 2.86e-06 3.50e-09 2.60e-06 2.58e-07 2.92e-01 4212 5.75e-08 0.27 4.94e-07 4.11e-10 4.41e-07 5.27e-08 1.16e-01 4213 4.76e-08 3.65 1.63e-07 5.01e-11 1.48e-07 1.50e-08 2.92e-01 4214 3.79e-09 -0.11 6.38e-08 1.23e-11 5.10e-08 1.28e-08 5.93e-02 3215 4.10e-09 3.35 1.41e-08 8.96e-13 1.27e-08 1.35e-09 2.91e-01 3216 4.03e-10 1.17 2.33e-09 1.03e-13 2.11e-09 2.21e-10 1.73e-01 3217 1.80e-10 1.72 6.17e-10 1.26e-14 5.59e-10 5.83e-11 2.91e-01 3218 5.44e-11 — 1.83e-10 1.56e-15 1.69e-10 1.48e-11 2.97e-01 3a.r. 1.83 1.95 3.00 1.94 2.02

robust convergence of almost second order

further investigation:higher-order splines (a posteriori estimator)convection-diffusion problems (stability issue)

Quadratic C1-spline collocation for reaction-diffusion ... filegeneral theory for spline-collocation...

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