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Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Polygonal Splines and Their Applications for Numerical Solution of PDE 1

Ming-Jun Lai

Department of Mathematics University of Georgia Athens, GA. U.S.A.

• Oct. 27, 2015

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

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

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Introduction

Multivariate splines, usually defined on a triangulation in 2D, or a tetrahedral partition in 3D, or spherical surface, or a simplicial partition in Rn, have been developed for 30 years and they are extremely useful to various numerical applications such as computer aided geometric design, numerical solutions of various linear and nonlinear partial differential equations, scattered data interpolation and fitting, image enhancements, spatial statistical analysis and data forecasting, and etc..

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Triangulated Splines for Applications

image data 2D spline interpolation2

• 2M. -J. Lai and L. L. Schumaker, Domain decomposition method for scattered

data fitting, SIAM J. Num. Anal. 47(2009) 911–928.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Triangulated Splines for Applications (II)

3

• 3M. -J. Lai and Meile, C. , Scattered data interpolation with nonnegative

preservation using bivariate splines, Computer Aided Geometric Design, vol. 34 (2015) pp. 37–49

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Triangulated Splines for Applications (III)

4

4Guo, W. H. and Lai, M. -J., Box Spline Wavelet Frames for Image Edge

Analysis, SIAM Journal Imaging Sciences, vol. 6 (2013) pp. 1553–1578.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Triangulated Splines for Applications (IV)

See 5

5Lai, 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 Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space 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 Sr

d(∆) = {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 Sr

d(∆) = {s ∈ C−1(Ω), s|ei ∈ Cri, ei ∈ E},

where E is the collection of interior edges of △. Each spline in Sr

d(△) has variable smoothness.

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

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Definition of Spline Functions

Let T = (x1, y1), (x2, y2), (x3, y3). For any point (x, y), let b1, 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!bi

1bj 2bk 3

which is called Bernstein-B´ ezier polynomials. For each T ∈ ∆, let S|T =

• i+j+k=d

cT

ijkBijk(x, y).

We use s = (cT

ijk, i + j + k = d, T ∈ ∆) be the coefficient vector to

denote a spline function in S−1

d (∆).

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Evaluation and Derivatives

We use the de Casteljau algorithm to evaluate a Bernstein-B´ ezier polynomial at any point inside the triangle. It is a simple and stable 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 directional derivatives.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Smoothness Condition between Triangles

Let T1 and T2 be two triangles in ∆ which share a common edge

• e. Then S ∈ Cr(T1 ∪ T2) if and only if the coefficients of cT1

ijk and

cT2

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]). We code them by Hc=0.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Integration

Let s be a spline in Sr

d(△) with

s|T =

i+j+k=d cT ijkBijk(x, y), T ∈ △. Then

• Ω

s(x, y)dxdy =

• T ∈△

AT d+2

2

• i+j+k=d

cT

ijk.

If p =

i+j+k=d aijkBijk(x, y) and q = i+j+k=d bijkBijk(x, y)

• ver 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 is a 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 Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Spline Approximation Order

We have (cf. [Lai and Schumaker’98]6) Theorem Suppose that △ is a β-quasi-uniform triangulation of domain Ω ∈ R2 and suppose that d ≥ 3r + 2. Fix 0 ≤ m ≤ d. Then for any f in a Sobolev space W m+1

p

(Ω), there exists a quasi-interpolatory spline Qf ∈ Sr

d(△) such that

f − Qfk,p,Ω ≤ C|△|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 Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space 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 ∈ Hr

0(Ω),

(1) where a(u, v) is the bilinear form defined by a(u, v) =     

• Ω

∇u · ∇vdxdy, k = 1;

• Ω

△u△vdxdy, k = 2, and f, v =

• Ω f(x, y)v(x, y)dxdy is the L2 inner product of f and
• v. Here Hk(Ω) and Hk

0 (Ω) are standard Sobolev spaces.

Clearly, our model problem is the Euler-Lagrange equation associated with a minimization prpblem: min

u∈Hk(Ω) E(u), u satisfies given boundary conditions

(2) where the energy functional E(u) = 1

2a(u, u) − f, u.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Our Spline Method

[1] We write s ∈ S−1

d (△) in

s(x, y)|t =

• i+j+k=dt

ct

i,j,kBt i,j,k(x, y),

(x, y) ∈ t ∈ △. Let c = (ct

i,j,k, i + j + k = dt, t ∈ △) be the B-coefficient vector

associated with s. [2] We compute mass and stiffness matrices: E(s) = 1 2c⊤Kc − c⊤Mf where K and M are diagonally block matrices. [3] Since s ∈ Sr

d(△), 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: min E(s), subject to Hc = 0, Bc = g.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Our Spline Method (II)

By using Lagrange multiplier method, we let L(c, α, β) = 1 2cT Kc − cT Mf + αHc + β(Bc − g) and compute local minimizers. That is, we solve   BT HT K B H     β α c   =   Mf g   (3) which can be solved using the least squares method or by an iterative algorithm in [Awanou, Lai, Wenston’068]. We minimize

• L(c, α, β) = 1

2cT Kc − cT Mf + αHc2 + βBc − g2.

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 Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

An Iterative Algorithm

We begin with

• K + 1

ǫ [B⊤, H⊤]⊤[B⊤H⊤]

• c(1) = Mf + 1

ǫ [B⊤, H⊤]⊤G Then we do the following iterative step

• K + 1

ǫ [B⊤, H⊤]⊤[B⊤H⊤]

• c(k+1) = Kc(k) + 1

ǫ [B⊤, H⊤]⊤G for k = 1, 2, · · · , 10 and ǫ > 0, e.g., ǫ = 10−6 with c = 0. Theorem (Awanou, Lai, and Wenston’06) Suppose that K is positive definite with respect to [B⊤H⊤]. Then the above iterative algorithm converges and c(k+1) − c ≤ Cǫk, ∀k ≥ 1.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

A Natural Question

A natural question follows: can we define splines on a partition containing not only triangles, but also other polygons, say over a Voronoi diagram (i.e. Dirichlet tessellation) or a patio? That is, can we be more versatile in solution of PDE? Can we be more efficient than the standard finite element method? These questions have been raised in the community of numerical solution of PDE. The so-called virtual element method and the discontinuous Galerkin method are developed recently.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Generalized Barycentric Coordinates

Similar to the barycentric coordinates b1, b2, b3 associated with a T = v1, v2, v3, on an arbitrary polygon P = v1, · · · , vn, say pentagon r ✁ ✁ ✁ ✁ r ❜ ❜ ❜ ❜ ❜ r ✓ ✓ ✓ ✓ r ❇ ❇ ❇ ❇ r there 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 Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

GBC functions

There are many kinds of GBC available. Construction of various GBC is pioneered by Wachspress in 1975. More and more GBC have been invented. Wachspress Coordinates, Mean Value Coordinates, Bilinear Coordinates, maximum entropy coordinates, discrete harmonic coordinates, Wachspress and mean value coordinates have 3D generalization 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, Acta

Numerica 24 (2015), 161–214.

• 10G. Manzini, A, Russo and N. Sukumar, New perspectives on polygonal and

polyhedral finite element methods, Mathematical Models and Methods in Applied Sciences Vol. 24, No. 8 (2014) 1665–1699.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space 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) n

j=1 wj(x),

i = 1, 2, · · · , n (5) which are called Wachspress barycentric coordinates. One can check 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 barycentric

coordinates for irregular polygons, J. Graph. Tools 7, 13–22.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Polygonal Bernstein-B´ ezier Functions

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

j (x) = d!

j!

n

• i=1

bji

i ,

|j| = d (6) be the polygonal Bernstein-B´ ezier Functions, where j = (j1, · · · , jn), ji, i = 1, · · · , n are nonnegative integers, j! = n

i=1 ji! and |j| = j1 + · · · + jn.

Let Φn,d be the linear space of functions of the form s(x) =

• |j|=d

cjBd

j (x),

x ∈ Pn (7) with real coefficients cj and let Sr

d(∆) = {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 Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Reduced Polygonal BB Functions

For convenience, we restrict ourselves to those BB functions in Φn,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 that they are able to reproduce all polynomials of degree d. Any such functions in Ψd(Pn) are called reduced Bernstein-B´ ezier functions. Thus, let us identify these functions in Ψd(Pn). Recall that any p ∈ Πd has a unique, d-variate blossom, P[p]. For p, its blossom P[p](x1, . . . , xd) with variables x1, . . . , xd ∈ R2 is uniquely defined by the three properties: (i) it is symmetric in the variables x1, . . . , xd; (ii) it is multi-affine, i.e., affine in each variable while the others are fixed; and (iii) it has the diagonal property, P[p](x, . . . , x

• d

) = p(x).

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Example 1

Consider d = 2 first. Let us first expand the x variable using the barycentric coordinates of x with respect to the vertex vi and its two 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 Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space 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 span{Fi, Li, i = 1, · · · , n} is able to reproduce Π2. Also, it is clear that each λi,j can be expressed by using bk, k = 1, · · · , n and hence, Fi, Li ∈ Φ2(Pn). These are reduced BB functions.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Reduced BB Functions of Order d ≥ 3

Let Fi = λd−1

i,0 , Fi,k =

d − 1 k

• φiλk

i,1λd−1−k i,0

+ d − 1 k − 1

• φi+1λd−k

i,−1λk−1 i,0 .

Theorem (Floater and Lai, 201512) For d ≥ 3 and for Wachspress coordinates, the reduced BB function space Ψd(Pn) is span{Fi, i = 1, . . . , n} ⊕ span{Fi,k, i = 1, . . . , n, k = 1, . . . , d − 1} ⊕ b W Πd−3, where Pn and b(x) = n

j=1 hj(x).

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

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Polygonal Spline Space

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

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 their common edge e or common vertex v if the intersection is not

• empty. We can define a continuous polygonal spline space of
• rder d ≥ 1 over Ω by

Sd(∆) = {s ∈ C(Ω), s|Pn ∈ Ψd(Pn), ∀Pn ∈ ∆}, (12) where for d = 1, we simply let Ψ1(Pn) = span{φi, i = 1, · · · , n} and for d ≥ 2, we use the reduced BB function space Ψd(Pn) discussed above.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space 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 on Wachspress coordinates. Then The dimension of Sd(∆) is dim(Sd(∆)) = #(V ) + (d − 1)#(E) + #(△)(d − 1)(d − 2)/2, (13) where V is the collection of all vertices of ∆, E is the collection of all edges of ∆ and #(∆) stands for the number of polygons in ∆. Open Question: How to construct Cr smooth spline functions

• ver a polygonal mesh ∆ for r ≥ 1?

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

An Extension to the Trivariate Setting

For simplicity, let us consider a hexahedron H ⊂ R3. It is a polyhedron with 6 faces, 12 edges and 8 vertices. Advantages: (1)H is a simple polyhedron, three incidental faces at each vertex

• f H.

(2) One can use such hexahedrons to partition any polyhedral domain in R3.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Hexahedral Partitions

Theorem We can use hexahadrons to partition any polyhedral domain Ω in R3. Proof. First for any tetrahedron T, one can decompose it into 4 hexahadrons. As one can use tetrahedrons to partition any polyhedral domain Ω ∈ R3, it follows that we can use hexadedrons to partition Ω.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Hexahedral Spline Space

Let ∆ be a collection of hexahedrons which partitions a polyhedral 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 the following Theorem For any fixed generalized barycentric coordinates, let Ψ2(H) be the space spanned by the reduced Bernstein-B´ ezier functions. Then the dimension of S2(∆) is dim(S2(∆)) = #(V ) + #(E), where V is the collection of all vertices of ∆ and E is the collection of all edges of ∆.

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Numerical Solution of PDE

We now present an application of these polygonal splines for numerical solution of Poisson equation with Dirichlet boundary value problem:

• −∆u = f,

x ∈ Ω ⊂ R2 u = g, x ∈ ∂Ω, (15) where Ω is a polygonal domain and ∆ is the standard Laplace

• perator. Let S2(∆) be the collection of polygonal spline functions
• f degree 2 over ∆. Let B(u, v) =
• Ω ∇u · ∇vdx be the standard

bilinear form associated with Poisson equation. Our numerical method is to solve the following weak solution: B(uh, v) = f, v, ∀v ∈ S2(∆) ∩ H1

0(Ω).

(16)

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space 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 and u3(x, y) = sin(π(x2 + y2)) + 1 to check the performance of our algorithm. 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 0.0631 64 225 1.1326e-04 8.6697e-04 0.0084 256 833 1.3895e-05 1.1069e-04 0.0013 1024 3201 1.6907e-06 1.4461e-05 1.5831e-04

Table : Convergence of serendipity quadratic finite elements over uniformly refined quadrangulations

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

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Uniform Refinement of Pentagonal Partitions

❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✟✟✟✟✟✟✟✟✟✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ v1 u1 v2 u2 v3 u3 v4 u4 v5 u5 p w1 w2 w3 w4 w5

• PPPP

P ❅ ❅ ❅ ✏ ✏ ✏ ✏ ✏ ˆ p ✄ ✄ ✄ ✄ ❈ ❈ ❈❈

• ❅

❅ ❅

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Partition of Domains

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

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Numerical Solution of PDE over Pentagonal Partitions

pentagons u1 − u1,h2 rates u2 − u2,h2 rates 6 1.0947e-03 1.0524e-02 36 8.6966e-05 3.65 8.9303e-04 3.55 216 7.4922e-06 3.53 7.5053e-05 3.57 1296 6.4614e-07 3.53 6.3632e-06 3.56 77760 5.9470e-08 3.44 5.5677e-07 3.51 pentagons u3 − u3,h2 rates 6 1.0979e-01 36 6.7861e-03 4.01 216 5.6689e-04 3.58 1296 5.0677e-05 3.48 77760 4.4986e-06 3.49

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

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

A Numerical Solution over Mixed Polygons

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Hexahedral Spline Approximation

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

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space 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 1.92 1.92 5.13e-01 ∆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 Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Conclusion and Future Research

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

Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Thanks for your attention!