Ming-Jun Lai Polygonal Splines and Their Applications for ...

Ming-Jun Lai

Motivation

PolynomialSplines

Spline Method forPDE

GBC

BB functions

Polygonal SplineSpace

HexahedralSpline Space

Solution of PDE

Polygonal Splines and Their Applicationsfor Numerical Solution of PDE 1

Ming-Jun Lai

Department of MathematicsUniversity of GeorgiaAthens, GA. U.S.A.

Oct. 27, 2015

based on a joint work with Michael Floater of University of Oslo, NorwayAtlanta, GA

1This research is supported by National Science Foundation #DMS 1521537

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Introduction

Multivariate splines, usually defined on a triangulation in 2D, or atetrahedral partition in 3D, or spherical surface, or a simplicialpartition in Rn, have been developed for 30 years and they areextremely useful to various numerical applications such ascomputer aided geometric design, numerical solutions of variouslinear and nonlinear partial differential equations, scattered datainterpolation and fitting, image enhancements, spatial statisticalanalysis and data forecasting, and etc..

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Triangulated Splines for Applications

image data 2D spline interpolation2

2M. -J. Lai and L. L. Schumaker, Domain decomposition method for scattereddata fitting, SIAM J. Num. Anal. 47(2009) 911–928.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Triangulated Splines for Applications (II)

3

3M. -J. Lai and Meile, C. , Scattered data interpolation with nonnegativepreservation using bivariate splines, Computer Aided Geometric Design, vol. 34(2015) pp. 37–49

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Triangulated Splines for Applications (III)

4

4Guo, W. H. and Lai, M. -J., Box Spline Wavelet Frames for Image EdgeAnalysis, SIAM Journal Imaging Sciences, vol. 6 (2013) pp. 1553–1578.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Triangulated Splines for Applications (IV)

See 55Lai, M. J., Shum, C. K., Baramidze, V. and Wenston, P., Triangulated Spherical

Splines for Geopotential Reconstruction, Journal of Geodesy, vol. 83 (2009) pp.695–708.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

What Are Multivariate Splines?

Let ∆ be a triangulation of a domain Ω ⊂ R2. For integers d ≥ 1,−1 ≤ r ≤ d define by

Srd(∆) = s ∈ Cr(Ω), s|t ∈ Pd, t ∈ ∆

the spline space of smoothness r and degree d over ∆.

In general, let r = (r1, · · · , rn) with ri ≥ 0 be a vector of integers.Define

Srd(∆) = s ∈ C−1(Ω), s|ei ∈ Cri , ei ∈ E,

where E is the collection of interior edges of 4. Each spline inSrd(4) has variable smoothness.

This can handle the situation of hanging nodes in a triangulation!

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Definition of Spline Functions

Let T = 〈(x1, y1), (x2, y2), (x3, y3)〉. For any point (x, y), letb1, b2, b3 be the solution of

x = b1x1 + b2x2 + b3x3

y = b1y1 + b2y2 + b3y3

1 = b1 + b2 + b3.

Fix a degree d > 0. For i+ j + k = d, let

Bijk(x, y) =d!

i!j!k!bi1b

j2b

k3

which is called Bernstein-Bezier polynomials.For each T ∈ ∆, let

S|T =∑

i+j+k=d

cTijkBijk(x, y).

We use s = (cTijk, i+ j + k = d, T ∈ ∆) be the coefficient vector todenote a spline function in S−1

d (∆).

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Evaluation and Derivatives

We use the de Casteljau algorithm to evaluate a Bernstein-Bezierpolynomial at any point inside the triangle. It is a simple andstable computation.Let T = 〈v1,v2,v3〉 and S|T =

∑i+j+k=d cijkBijk(x, y). Then

directional derivative

Dv2−v1S|T = d∑

i+j+k=d−1

(ci,j+1,k − ci+1,j,k)Bijk(x, y).

Similar for Dv3−v1S|T .Dx and Dy are linearly combinations of these two directionalderivatives.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Smoothness Condition between Triangles

Let T1 and T2 be two triangles in ∆ which share a common edgee. Then S ∈ Cr(T1 ∪ T2) if and only if the coefficients of cT1

ijk andcT2

ijk satisfy the following linear conditions. E.g.,

S ∈ C0(T1 ∪ T2) iff cT1

0,j,k = cT2

j,k,0, j + k = d

S ∈ C1(T1 ∪ T2) iff cT1

1,j,k = b1cT2

j+1,k,0 + b2cT2

j,k+1,0 + b3cT2

j,k,1

for i+ k = d− 1 and etc. (cf. [Farin, 86] and [de Boor, 87]). Wecode them by Hc=0.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Integration

Let s be a spline in Srd(4) with

s|T =∑

i+j+k=d cTijkBijk(x, y), T ∈ 4. Then∫

Ω

s(x, y)dxdy =∑T∈4

AT(d+2

2

) ∑i+j+k=d

cTijk.

If p =∑

i+j+k=d aijkBijk(x, y) and q =∑

i+j+k=d bijkBijk(x, y)over a triangle T , then∫

T

p(x, y)q(x, y)dxdy = a>Mdb,

where a = (aijk, i+ j + k = d)>, b = (bijk, i+ j + k = d)>, Md isa symmetric matrix with known entries (cf. [Chui and Lai, 1992]).Similarly, we have (cf. [Awanou and Lai, 2005])∫

T

p(x, y)q(x, y)r(x, y)dxdy = a>Adb c.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Spline Approximation Order

We have (cf. [Lai and Schumaker’98]6)

Theorem

Suppose that 4 is a β-quasi-uniform triangulation of domainΩ ∈ R2 and suppose that d ≥ 3r + 2. Fix 0 ≤ m ≤ d. Then for anyf in a Sobolev space Wm+1

p (Ω), there exists a quasi-interpolatoryspline Qf ∈ Sr

d(4) such that

‖f −Qf‖k,p,Ω ≤ C|4|m+1−k|f |d+1,p,Ω,∀0 ≤ k ≤ m+ 1,

for a constant C > 0 independent of f , but dependent on β and d.

See more detail in monograph7

6Lai, M. J. and Schumaker, L. L., Approximation Power of Bivariate Splines,Advances in Computational Mathematics, vol. 9 (1998) pp. 251–279.

7M. -J. Lai and L. L. Schumaker, Spline Functions on Triangulations,Cambridge University Press, 2007.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

The Weak Form of PDE’s

The weak formulation for elliptic PDE’s reads : find u ∈ Hk(Ω)which satisfies its boundary condition such that

a(u, v) = 〈f, v〉, ∀v ∈ Hr0 (Ω), (1)

where a(u, v) is the bilinear form defined by

a(u, v) =

∫

Ω

∇u · ∇vdxdy, k = 1;∫Ω

4u4vdxdy, k = 2,

and 〈f, v〉 =∫

Ωf(x, y)v(x, y)dxdy is the L2 inner product of f and

v. Here Hk(Ω) and Hk0 (Ω) are standard Sobolev spaces.

Clearly, our model problem is the Euler-Lagrange equationassociated with a minimization prpblem:

minu∈Hk(Ω)

E(u), u satisfies given boundary conditions (2)

where the energy functional E(u) = 12a(u, u)− 〈f, u〉.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Our Spline Method

[1] We write s ∈ S−1d (4) in

s(x, y)|t =∑

i+j+k=dt

cti,j,kBti,j,k(x, y), (x, y) ∈ t ∈ 4.

Let c = (cti,j,k, i+ j + k = dt, t ∈ 4) be the B-coefficient vectorassociated with s.[2] We compute mass and stiffness matrices:

E(s) =1

2c>Kc− c>M f

where K and M are diagonally block matrices.[3] Since s ∈ Sr

d(4), we have the smoothness conditions Hc = 0.Also, the boundary condition can be written as Bc = g.[4] FInally we solve the constrained minimization problem:

minE(s), subject to Hc = 0, Bc = g.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Our Spline Method (II)

By using Lagrange multiplier method, we let

L(c, α, β) =1

2cTKc− cTMf + αHc + β(Bc− g)

and compute local minimizers. That is, we solve BT HT K0 0 B0 0 H

βαc

=

M fg0

(3)

which can be solved using the least squares method or by aniterative algorithm in [Awanou, Lai, Wenston’068].

We minimizeL(c, α, β) = 1

2cTKc− cTM f + α‖Hc‖2 + β‖Bc− g‖2.8Awanou, G., Lai, M. J. and Wenston, P., The Multivariate Spline Method for

Scattered Data Fitting and Numerical Solution of Partial Differential Equations,Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. J.pp. 24–74

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

An Iterative Algorithm

We begin with(K +

1

ε[B>, H>]>[B>H>]

)c(1) = M f +

1

ε[B>, H>]>G

Then we do the following iterative step(K +

1

ε[B>, H>]>[B>H>]

)c(k+1) = Kc(k) +

1

ε[B>, H>]>G

for k = 1, 2, · · · , 10 and ε > 0, e.g., ε = 10−6 with c = 0.

Theorem (Awanou, Lai, and Wenston’06)

Suppose that K is positive definite with respect to [B>H>]. Thenthe above iterative algorithm converges and

‖c(k+1) − c‖ ≤ Cεk, ∀k ≥ 1.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

A Natural Question

A natural question follows: can we define splines on a partitioncontaining not only triangles, but also other polygons, say over aVoronoi diagram (i.e. Dirichlet tessellation) or a patio?

That is, can we be more versatile in solution of PDE? Can we bemore efficient than the standard finite element method?

These questions have been raised in the community of numericalsolution of PDE. The so-called virtual element method and thediscontinuous Galerkin method are developed recently.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Generalized Barycentric Coordinates

Similar to the barycentric coordinates b1, b2, b3 associated with aT = 〈v1,v2,v3〉, on an arbitrary polygon P = 〈v1, · · · ,vn〉, saypentagon

r

rbb

bbb

r

rBBBB

rthere are functions b1 ≥ 0, · · · , bn ≥ 0

satisfying

n∑i=1

bi(x) = 1 and x =n∑

i=1

bi(x)vi ∀x ∈ R2 (4)

which are called generalized barycentric coordinates (GBC).

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

GBC functions

There are many kinds of GBC available. Construction of variousGBC is pioneered by Wachspress in 1975. More and more GBChave been invented.

Wachspress Coordinates,Mean Value Coordinates,Bilinear Coordinates,maximum entropy coordinates,discrete harmonic coordinates,Wachspress and mean value coordinates have 3Dgeneralization and been extended in the spherical setting.

We refer to [Floater, 20159] and [G. Manzini, A. Russo and N.Sukumar 201410] for a summary surveying various types of GBC.

9M. Floater, Generalized barycentric coordinates and applications, ActaNumerica 24 (2015), 161–214.

10G. Manzini, A, Russo and N. Sukumar, New perspectives on polygonal andpolyhedral finite element methods, Mathematical Models and Methods in AppliedSciences Vol. 24, No. 8 (2014) 1665–1699.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Wachspress Coordinates

Example (Wachspress coordinates)

For any polygon Pn with n sides, let

Ai(x) = A(x,vi,vi+1), Ci = A(vi−1,vi,vi+1)

be the signed area of triangles 〈x,vi,vi+1〉 and 〈vi−1,vi,vi+1〉,respectively. Setting wi(x) = Ci/(Ai−1(x)Ai(x)) we define

φi(x) =wi(x)∑nj=1 wj(x)

, i = 1, 2, · · · , n (5)

which are called Wachspress barycentric coordinates. One cancheck these φi satisfy (4).These coordinates have been developed in [Wachspress, 1975],[Warren, 1996], ... . The above form was given in [Meyer et al,200211]

11M. Meyer, A. Barr, H. Lee and M. Desbrun (2002), Generalized barycentriccoordinates for irregular polygons, J. Graph. Tools 7, 13–22.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Polygonal Bernstein-Bezier Functions

For each n-gon Pn and for any GBC b1, · · · , bn, we let

Bdj (x) =

d!

j!

n∏i=1

bjii , |j| = d (6)

be the polygonal Bernstein-Bezier Functions, wherej = (j1, · · · , jn), ji, i = 1, · · · , n are nonnegative integers,j! =

∏ni=1 ji! and |j| = j1 + · · ·+ jn.

Let Φn,d be the linear space of functions of the form

s(x) =∑|j|=d

cjBdj (x), x ∈ Pn (7)

with real coefficients cj and let

Srd(∆) = s ∈ Cr(Ω) : s|Pn∈ Φn,d, Pn ∈∆. (8)

be the polygonal spline space of smoothness r and degree d,where ∆ is a collection of polygons.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Reduced Polygonal BB Functions

For convenience, we restrict ourselves to those BB functions inΦn,d(P ) which enable us to reproduce polynomials of degree d.Let Ψd(Pn) be the collection of all those terms in Φd(Pn) such thatthey are able to reproduce all polynomials of degree d. Any suchfunctions in Ψd(Pn) are called reduced Bernstein-Bezierfunctions.

Thus, let us identify these functions in Ψd(Pn).

Recall that any p ∈ Πd has a unique, d-variate blossom, P[p]. Forp, its blossom P[p](x1, . . . ,xd) with variables x1, . . . ,xd ∈ R2 isuniquely defined by the three properties: (i) it is symmetric in thevariables x1, . . . ,xd; (ii) it is multi-affine, i.e., affine in eachvariable while the others are fixed; and (iii) it has the diagonalproperty,

P[p](x, . . . ,x︸︷︷︸d

) = p(x).

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Example 1

Consider d = 2 first. Let us first expand the x variable using thebarycentric coordinates of x with respect to the vertex vi and itstwo neighbors, i.e., with respect to Ti := [vi−1,vi,vi+1]. That is,

x =

1∑j=−1

λi,j(x)vi+j , 1 =

1∑j=−1

λi,j(x). (9)

By the properties (4) and (9), we have

p(x) =P[p](x,x) =

n∑i=1

bi(x)P[p](vi,x)

=

n∑i=1

bi(x)1∑

j=−1

λi,j(x)P[p](vi,vi+j).

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Example 1(Cont.)

After a simplification, we have

p(x) =

n∑i=1

bi(x)λi,0(x)p(vi)+

n∑i=1

P[p](vi,vi+j)(bi(x)λi,1(x) + bi+1(x)λi+1,−1(x)).(10)

Thus, we are ready to define

Fi(x) = bi(x)λi,0(x) and Li = bi(x)λi,1(x) + bi+1(x)λi+1,−1(x).(11)

We know that spanFi, Li, i = 1, · · · , n is able to reproduce Π2.Also, it is clear that each λi,j can be expressed by usingbk, k = 1, · · · , n and hence, Fi, Li ∈ Φ2(Pn). These are reducedBB functions.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Reduced BB Functions of Order d ≥ 3

Let

Fi = λd−1i,0 , Fi,k =

(d− 1

k

)φiλ

ki,1λ

d−1−ki,0 +

(d− 1

k − 1

)φi+1λ

d−ki,−1λ

k−1i,0 .

Theorem (Floater and Lai, 201512)

For d ≥ 3 and for Wachspress coordinates, the reduced BBfunction space Ψd(Pn) is

spanFi, i = 1, . . . , n ⊕ spanFi,k, i = 1, . . . , n, k = 1, . . . , d− 1

⊕ b

WΠd−3,

where Pn and b(x) =∏n

j=1 hj(x).

12M. Floater and M. -J. Lai, Polygonal slpine spaces and the numerical solutionof the Poisson equation, submitted, (2015)

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Polygonal Spline Space

Let ∆ be a collection of polygons with arbitrary number sides. LetΩ =

⋃P∈∆ P be the domain consisting of these polygons in ∆.

We assume that the interiors of any two polygons P and P from∆ do not intersect and the intersection of P and P is either theircommon edge e or common vertex v if the intersection is notempty. We can define a continuous polygonal spline space oforder d ≥ 1 over Ω by

Sd(∆) = s ∈ C(Ω), s|Pn∈ Ψd(Pn),∀Pn ∈∆, (12)

where for d = 1, we simply let Ψ1(Pn) = spanφi, i = 1, · · · , nand for d ≥ 2, we use the reduced BB function space Ψd(Pn)discussed above.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Dimension of Polygonal Spline Space (d ≥ 2)

We can establish the dimension of Sd(∆).

Theorem

Suppose that Sd(∆) is a polygonal spline space based onWachspress coordinates. Then The dimension of Sd(∆) is

dim(Sd(∆)) = #(V ) + (d− 1)#(E) + #(4)(d− 1)(d− 2)/2, (13)

where V is the collection of all vertices of ∆, E is the collection ofall edges of ∆ and #(∆) stands for the number of polygons in ∆.

Open Question: How to construct Cr smooth spline functionsover a polygonal mesh ∆ for r ≥ 1?

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

An Extension to the Trivariate Setting

For simplicity, let us consider a hexahedron H ⊂ R3. It is apolyhedron with 6 faces, 12 edges and 8 vertices.

Advantages:(1)H is a simple polyhedron, three incidental faces at each vertexof H.(2) One can use such hexahedrons to partition any polyhedraldomain in R3.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Hexahedral Partitions

Theorem

We can use hexahadrons to partition any polyhedral domain Ω inR3.

Proof.

First for any tetrahedron T , one can decompose it into 4hexahadrons.

As one can use tetrahedrons to partition any polyhedral domainΩ ∈ R3, it follows that we can use hexadedrons to partition Ω.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Hexahedral Spline Space

Let ∆ be a collection of hexahedrons which partitions apolyhedral domain Ω ∈ R3. Let

Sd(∆) = s ∈ C(Ω), s|H ∈ Ψd(Pn),∀H ∈∆, (14)

where Ψd(H) is the reduced BB function space for d ≥ 1.With a straightforward extension, we are able to prove thefollowing

Theorem

For any fixed generalized barycentric coordinates, let Ψ2(H) bethe space spanned by the reduced Bernstein-Bezier functions.Then the dimension of S2(∆) is

dim(S2(∆)) = #(V ) + #(E),

where V is the collection of all vertices of ∆ and E is thecollection of all edges of ∆.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Numerical Solution of PDE

We now present an application of these polygonal splines fornumerical solution of Poisson equation with Dirichlet boundaryvalue problem:

−∆u = f, x ∈ Ω ⊂ R2

u = g, x ∈ ∂Ω,(15)

where Ω is a polygonal domain and ∆ is the standard Laplaceoperator. Let S2(∆) be the collection of polygonal spline functionsof degree 2 over ∆. Let B(u, v) =

∫Ω∇u · ∇vdx be the standard

bilinear form associated with Poisson equation. Our numericalmethod is to solve the following weak solution:

B(uh, v) = 〈f, v〉,∀v ∈ S2(∆) ∩H10 (Ω). (16)

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Numerical Solutions over Quadrilaterals

We have done several tests on smooth solutions:u1(x, y) = 1/(1 + x+ y), u2(x, y) = x4 + y4 andu3(x, y) = sin(π(x2 + y2)) + 1 to check the performance of ouralgorithm.

Quads #DoFs ‖u1 − u1,h‖∞ ‖u2 − u2,h‖∞ ‖u3 − u3,h‖∞4 21 0.0084 0.0458 0.3218

16 65 8.5628e-04 0.0065 0.063164 225 1.1326e-04 8.6697e-04 0.0084

256 833 1.3895e-05 1.1069e-04 0.00131024 3201 1.6907e-06 1.4461e-05 1.5831e-04

Table : Convergence of serendipity quadratic finite elements overuniformly refined quadrangulations

The convergence rate is 3, consistent with the theory and thenumerical results from [Rand, Gillette and Babaj, 2014].

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Uniform Refinement of Pentagonal Partitions

AAAAAAAAAA

HHHHH

HHH

HH

v1 u1v2

u2

v3

u3

v4

u4

v5

u5

p

w1

w2

w3w4

w5

PPPPP

@@@

p

CCCC

@@@

Figure : An illustration of uniform pentagon refinement with verticesvi,ui,wi, i = 1, · · · , 5

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Partition of Domains

Figure : A Pentagon Partition (top-left) and its refinements

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Numerical Solution of PDE over PentagonalPartitions

pentagons ‖u1 − u1,h‖2 rates ‖u2 − u2,h‖2 rates6 1.0947e-03 0 1.0524e-02 036 8.6966e-05 3.65 8.9303e-04 3.55

216 7.4922e-06 3.53 7.5053e-05 3.571296 6.4614e-07 3.53 6.3632e-06 3.56

77760 5.9470e-08 3.44 5.5677e-07 3.51pentagons ‖u3 − u3,h‖2 rates

6 1.0979e-01 036 6.7861e-03 4.01

216 5.6689e-04 3.581296 5.0677e-05 3.48

77760 4.4986e-06 3.49

Table : Convergence of C0 quadratic serendipity finite elements overuniformly refined pentagon partitions in Fig. 2

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

A Numerical Solution over Mixed Polygons

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Hexahedral Spline Approximation

Recall the definition of Fi associated with vertex vi of ahexahedron H and Li associated with edge ei of H. For acollection ∆ of hexahedraons, we can define a locally suportedpolyhedral spline function Fv associated with each vertex v of ∆and Le associated with each edge.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Hexahedral Spline Approximation (II)

We define a quasi-interpolatory operator

Q(f) =∑v∈∆

f(v)Fv +∑e∈∆

f(ue)Le

where ue is the midpoint of the edge e. Then we have

Theorem

For all quadratic polynomial p in total degree,

Q(p) = p.

Qf − f Rate Dx Rate Dy Rate Dz Rate∆ 4.92e-01 0 1.92 0 1.92 0 5.13e-01 0∆1 6.29e-02 2.96 4.86e-01 1.98 4.86e-01 1.98 1.25e-01 2.03∆2 8.02e-03 2.97 1.23e-01 1.98 1.23e-01 1.98 3.11e-02 2.0

Table : Convergence C0 quadratic hexahedral splines to x3 + y3 + z3

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Conclusion and Future Research

There are a lot of research remaining open:

dimensions of polygonal BB functions, dimension ofpolygonal spline space for other GBC’s d ≥ 3 thanWachspress coordinates.how to define Cr smooth spline spaces with r ≥ 1, what arethe approximation power of those spline spaces.Why is the approximation rate based on pentagonalrefinement scheme faster?Refinability: let ∆2 be a uniform refinement of ∆. IsSd(∆) ⊂ Sd(∆2)?We are extending our study to the 3D setting , in particular,numerical solution of PDE.

Ming-Jun Lai

Motivation

PolynomialSplines


GBC

BB functions



Solution of PDE

Thanks for your attention!

Ming-Jun Lai Polygonal Splines and Their Applications for ...

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