Finite Element Solutions for Three Dimensional Elliptic Boundary ...

Finite Element Solutions for Three Dimensional Elliptic Boundary Value Problems on Unbounded Domains Hae-Soo Oh1 2 Dept. of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 282230001 Jae-Heon Yun5 Dept. of Mathematics, Institute for Basic Sciences & College of Natural Sciences Chungbuk National University, Korea Bong Soo Jang Dept. of Mathematics, Kent State University, Ashtabula, Ohio 44004

The finite element(FE) solutions of a general elliptic equation −div([aij ] · ∇u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R3 , is considered. The most common approach to deal with exterior domain problems is truncating an unbounded ¯ ∞ is bounded, and imposing an artificial subdomain Ω∞ , so that the remaining part ΩB = Ω\Ω ¯∞ ∩ Ω ¯ B . In this paper, instead of boundary condition on the resulted artificial boundary Γa = Ω discarding an unbounded subdomain Ω∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babuˇska and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This c ??? John Wiley method thus does have neither artificial boundary nor any restrictions on f . ° & Sons, Inc. Keywords: Method of Auxiliary Mapping, the p-Version of the Finite Element Method, Weighted Ritz-Galerkin Method, Weighted Sobolev Space, Infinite Elements.

1

Correspondence to: Hae-Soo Oh, Dept. of Mathematics, Univ. of North Carolina at Charlotte, Charlotte, NC 28223(e-mail:[email protected]) 2 The research of this author is supported in part by NSF grant INT-9910345 and funds provided by the Univ. of NC at Charlotte. 5 This research is supported in part by KOSEF grant 976-0100-002-2. Numerical Methods for Partial Differential Equations ???, 1 22 (???)

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OH, YUN AND JANG

I. INTRODUCTION In this paper, we are concerned with elliptic boundary value problems on an exterior domain, the complement of a region which is simply connected, closed, and bounded in R3 . This paper is a three dimensional (3-D) extension of ([24], [25]) dealt with two dimensional (2-D) unbounded domain problems. Related to the finite element method(FEM), much work has been done on the unbounded domain problems ([1], [10], [11], [12], [13], [15], [16], [19], [21]). Several numerical methods for unbounded domain problems were suggested and the following approaches are typical: domain truncating method, coupling boundary element method with FEM ([17]), and using infinite elements ([6], [7], [8], [14], [28]). For domain truncating method, the exterior domain Ω is divided into two subdomains, ΩB = {x ∈ Ω : kxk < c} and Ω∞ = R3 \ΩB . Discarding the unbounded subdomain Ω∞ and imposing a reasonable artificial boundary condition (for example, Sommerfeld-type ¯∞ ∩ Ω ¯ B is the essence of this boundary condition) along the artificial boundary Γa = Ω method. The accuracy of this method generally depends on the choice of Ω B as well as an artificial boundary condition. On the other hand, the infinite element methods are partitioning Ω∞ into a finite number of infinite elements and are constructing special basis functions, which decay as kxk → ∞, for the shape functions on the infinite elements. This method uses non-conventional basis functions and hence non-standard finite element codes are required. The proposed method in this paper for unbounded domain problems is similar to the infinite element approach in principle. However, the proposed method uses the conventional finite element method on Ω∞ by transforming it into a bounded domain ˆ ∞ . The method is schematically shown in Fig. 1. Our method uses the conventional Ω ˆ ∞ , in which two bounded domains are finite element method on a bounded domain ΩB ∪ Ω hypothetically connected through an auxiliary mapping ϕ∞ . Neither artificial boundary conditions nor special basis functions are necessary for this method. Unlike the domain truncating method, the accuracy of finite element solutions by this approach virtually does not depend on the size of the bounded domain ΩB . Moreover, our method has no restriction on the support of the source function f . However, the transformed bilinear ˆ ∞ is nonsymmetric, nevertheless the bilinear form on ΩB is the standard form on Ω symmetric one. This paper is organized as follows: in section II, weight functions are introduced. Weighted Galerkin method as well as the existence of the solution of the weighted variational equation are stated. In section III, the proposed mapping method for the 3-D unbounded domain problems is presented. In section IV, in order to demonstrate the effectiveness of our mapping method, various numerical results by our method are compared with those obtained by the conventional methods. Finally, in appendix, the construction of weight functions and the change of variables related to the auxiliary mapping used for our method are stated and formulated.

II. THE WEIGHTED GALERKIN METHOD Throughout this paper, Ω ⊂ R3 denotes an unbounded domain which is the exterior of a closed simply connected bounded domain B and hence ∂Ω = ∂B(see, Fig. 1). The

ELLIPTIC PROBLEMS ON 3-D UNBOUNDED DOMAINS

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rectangularp coordinates of a point x ∈ Ω are denoted by either (x1 , x2 , x3 ) or (x, y, z), and kxk = x21 + x22 + x23 . ∂u ∂u ∂u T , , ) , It is convenient to employ the operators ∇, div, and ∆, defined by ∇x u = ( ∂x1 ∂x2 ∂x3 P3 ∂fi div(f ) = i=1 , where f = (f1 , f2 , f3 )T , and ∆u = div (∇x u). Here T means trans∂xi pose. Let g : Ω −→ [0, 1] be a smooth cutoff function such that ½ 0 for kxk ≤ c, g(x) = 1 for kxk > c + b. There are several different forms of cut-off functions. The one used in this paper is constructed in appendix. The constant c indicates the radius of the smallest sphere which contains a selected bounded sub-domain, and the constant b is specified later so that the norm of the gradient vector of the weight function e−2µg(x)kxk can be small. The symbol L2 (Ω; µ, g) stands for the set of all functions, u, that satisfy o1/2 nZ < ∞. (2.1) e−2µg(x)kxk |u|2 dx kukL2 (Ω;µ,g) = Ω

1

A Hilbert space W (Ω; µ, g), to which the generalized solution of our model problem belongs, is the set of all functions u satisfying kukW 1 (Ω;µ,g) < ∞, where 3 o1/2 n X ∂u 2 . kL2 (Ω;µ,g) k kukW 1 (Ω;µ,g) = kuk2L2 (Ω;µ,g) + ∂xi i=1

(2.2)

Here 0 ≤ µ ¿ 1 is a real number which gives the damping effects of the weight function. In particular, W 0 (Ω; µ, g) = L2 (Ω; µ, g). For brevity, in this paper, we are concerned with elliptic boundary value problems. However this method can be easily extended for elasticity problems on unbounded domains (see [24]). Consider a model elliptic boundary value problem, −div([aij ] · ∇u(x)) + u(x) = f (x) in Ω,

(2.3)

u(x) = 0 on ∂Ω, u goes to zero as kxk → ∞,

(2.4) (2.5)

where ∂Ω is Lipschitz continuous, f ∈ L2 (Ω; µ, g) decays, but may not have compact support, and the coefficient matrix is symmetric and positive definite at each point x ∈ Ω : 3 X

i,j=1 3 X

i,j=1

aij (x)ηi ηj ≥ α aij (x)ηj ξi ≤ β

3 X

ηi2 ,

(2.6)

i=1

3 ³X i=1

ξi2

3 ´1/2 ³X j=1

ηj2

´1/2

,

(2.7)

for all triples of real numbers (η1 , η2 , η3 ), (ξ1 , ξ2 , ξ3 ). Here β ≥ α > 0 are constants. Let us note that f (x) may not have compact support. However, in most physics and engineering problems on exterior domain, the right side function f (x) either has compact support or tends to decay quickly. Numerical tests show that our method yields highly accurate FE solutions to the elliptic problem (2.3)-(2.5) with rapidly decaying f (x).

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OH, YUN AND JANG

It was proved in ([25]) that kukL2 (Ω;µ,g) ≤ Ckf kL2 (Ω;µ,g) , where C is a constant independent of u and f . Thus, if f decays, then so does u, and for each f ∈ L2 (Ω; µ, g) that decays, the problem has a unique solution. Hence the elliptic problem (2.3)-(2.5) is well posed. Next, consider the variational equation for this problem. We define the Weighted Residue Method corresponding to (2.3)-(2.5) as follows: Find u ∈ V such that for all v ∈ V, Z n ³ ´ o −div [aij ] · ∇x u(x) + u(x) − f (x) v(x)e−2µg(x)kxk dx = 0, (2.8) Ω

where V = W01 (Ω; µ, g) = {v ∈ W 1 (Ω; µ, g) : v = 0 along ∂Ω} . If µ ≈ 1, then the data on Ω∞ make little contributions. On the other hand, if µ is too small, then the data coming from Ω∞ are overwhelming. The size of µ depends on the size of ΩB . In other words, it is selected so that the weight function starts to damp at the far outside of Ω B . µ is usually selected to be about 0.05. Applying the divergence theorem gives Z ³ ´ div [aij (x)] · ∇u(x) v(x)e−2µg(x)kxk dx − ZΩ = e−2µg(x)kxk (∇u(x))T · [aij (x)] · (∇v(x))dx Ω Z v(x)(∇u(x))T · [aij (x)] · (∇e−2µg(x)kxk )dx + Ω Z ν T · [aij (x)] · (∇u(x))v(x)e−2µg(x)kxk ds, − ∂Ω

where ν(x) = (ν1 (x), ν2 (x), ν3 (x))T is the outward unit normal vector to the boundary ∂Ω. Thus, the Weighted Residue Method ([9], [26], [27]) can be restated as follows: Find u∈V A(u, v) = F(v) where A(u, v) =

Z n Ω

for all v ∈ V,

(2.9)

e−2µg(x)kxk (∇u(x))T · [aij ] · (∇v(x))

+ v(x)(∇u(x))T · [aij ] · (∇e−2µg(x)kxk ) o + e−2µg(x)kxk u(x)v(x) dx, Z e−2µg(x)kxk f (x)v(x)dx. F(v) =

(2.10) (2.11)

Ω

Since g(x) = 0 for kxk < c, the variational formulation on ΩB is the standard one. That is, (2.10)-(2.11) can be rewritten as follows: Z Z f (x)v(x)dx, [(∇x u(x))T · [aij ] · (∇x v(x)) + u(x)v(x)]dx and F(v) = A(u, v) = ΩB

ΩB

ELLIPTIC PROBLEMS ON 3-D UNBOUNDED DOMAINS

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which is the standard variational formulation of (2.3)-(2.4). The weight function (e−2µg(x)kxk ) is 1 when kxk < c and it is e−2µkxk when kxk > c+b. Hence the gradient of the weight function is independent of ∇x g(x) when kxk is not in [c, c + b]. However, when kxk is in [c, c + b], the gradient vector of the weight function depends on ∇x g(x) as follows ∇x e−2µg(x)kxk = −2µe−2µg(x)kxk ∇(g(x)kxk) = −2µe−2µg(x)kxk {h(kxk − c) + kxk

dh (kxk − c)}∇kxk. dt

Here h(t) is defined by (A1) in appendix A. Moreover, ρ, defined by ρ = max {h(t) + (t + c)

dh(t) : 0 ≤ t ≤ b}, dt

(2.12)

plays a key role in proving the following existence theorem in ([25]), for the weighted variational problem (2.9). Theorem 2.1. Suppose the constants µ and ρ are properly selected so that 0 ≤ µρ < α . Then the variational problem (2.9) has a unique solution, uex (x) in W01 (Ω; µ, g). β

III. FINITE ELEMENT SPACES FOR 3-D UNBOUNDED DOMAIN PROBLEMS In this section, the Method of Auxiliary Mapping ([2], [20], [22], [23]), developed by Babuˇska and Oh, is modified for unbounded domain problems. Let S1 = {ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 : k(ξ1 , ξ2 , ξ3 )k ≤ 1} and S∞ = Ω ∩ {x ∈ R3 : kxk ≥ c}. Then the inversion mapping ϕ∞ : S∞ → S1 is defined by ϕ∞ (x1 , x2 , x3 ) = (

cx1 cx2 −cx3 , , ), kxk2 kxk2 kxk2

c > 0.

(3.1)

In the following, the transformed subsets and the transformed functions by the inversion ˆ = h ◦ ϕ−1 respectively. mapping is denoted by Sˆ = ϕ∞ (S) and h ∞ Then the essential components of our method are as follows: 1. The given exterior domain is divided into two regions; a bounded subregion Ω B and ¯ B . Actually, for computational advantages, Ω an unbounded subregion Ω∞ = Ω \ Ω is decomposed into two parts so that ΩB ∩ Ω∞ can be a polyhedron with triangular faces (see Fig. 1). In other words, the outer boundary of ΩB is a polyhedron inscribed in the sphere with radius c (see Fig. 2). ˆ ∞ (slightly 2. The unbounded subregion Ω∞ is mapped to another bounded region Ω larger than a unit ball since ∂Ω∞ is a polyhedron inscribed in the sphere) by the inversion mapping ϕ∞ . ˆ ∞ in which 3. The standard p-version of FEM is applied to the bounded region ΩB ∪ Ω ˆ Ω∞ is (fictitiously) connected to ΩB through the auxiliary mapping ϕ∞ . It has no ˆ ∞ ∪ΩB for the Finite Element Method, the artificial boundaries. In triangulating Ω ˆ ∞ share faces and nodes along ΩB ∩Ω∞ and the corresponding faces and nodes on Ω the same node numbers. Thus, two regions are treated as one connected domain in the computation process (FEM codes). Moreover, the transformed bilinear form,

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T Triangular Face :

S

q

B ΩB

Ω

T

S

Ω =R

3

B

Ω

FIG. 1. Schematic Diagram for 3-D exterior domain Ω and Mapped unbounded subdomain ˆ ∞ = ϕ∞ (Ω∞ ). The left dotted circle represents a sphere of radius c and the right dotted circle Ω represents a sphere of radius 1. An actual Tˆ∞ is shown in Fig. 3.

given in Lemma 3.1, is used for computing local stiffness matrices of the elements ˆ ∞. in the transformed domain Ω The novelty of our method is to obtain an accurate numerical solution without introducing any artificial boundaries. A. The Change of Variables

The Jacobian of the auxiliary mapping ϕ∞ is

J(ϕ∞ ) =

µ

c kxk4

¶



 −x21 + x22 + x23 −2x1 x2 2x1 x3  . −2x1 x2 x21 − x22 + x23 2x2 x3 2 2 2 −2x1 x3 −2x2 x3 −(x1 + x2 − x3 )

and hence its determinant is |J(ϕ∞ )| = |J(ϕ−1 ∞ )| =

c3 −1 −1 . Since J(ϕ∞ ) ◦ ϕ−1 , we have ∞ = [J(ϕ∞ )] kxk6

c3 , which is singular at the origin. kξk6

ELLIPTIC PROBLEMS ON 3-D UNBOUNDED DOMAINS

7

ˆ ·) defined on the mapped bounded domain Let us consider the bilinear form A(·, ˆ Ω∞ = ϕ∞ (Ω∞ ), which is the transformation of A(·, ·) under the auxiliary mapping. In ∂ ∂ ∂ T , ) and ξ = (ξ, η, ζ). what follows, ∇ξ = ( , ∂ξ ∂η ∂ζ By using Lemma 1.1 of appendix B, we have the following lemma which shows that the ˆ ˆ (ˆ ˆ(ξ)), would be infinite, unless the weight function transformed bilinear form, A| Ω∞ u(ξ), v is employed. −1 −1 T −1 Lemma 3.1. Let |J(ϕ−1 } · [aij ] · [J(ϕ−1 = [qij ]. Then, for u, v ∈ ∞ )|{[J(ϕ∞ )] ∞ )] 1 ˆ v )| ˆ , where ˆ u, vˆ)|Ωˆ ∞ and F(v)|Ω∞ = F(ˆ W (Ω; µ, g), we have A(u, v)|Ω∞ = A(ˆ Ω∞ Z £ ¤ ˆ u, vˆ)| ˆ := k1 (ξ) (∇ξ u ˆ)T · [qij ] · (∇ξ vˆ) dξ A(ˆ Ω∞ ˆ∞ Ω Z £ ¤ k2 (ξ)ˆ v (ξ) (∇ξ u ˆ)T · [qij ] · (ξ) dξ + ˆ Z Ω∞ + k3 (ξ)ˆ u(ξ)ˆ v (ξ)dξ, (3.2) ˆ∞ Ω Z ˆ v )| ˆ := k3 (ξ)fˆ(ξ)ˆ v (ξ)dξ, (3.3) F(ˆ Ω∞ ˆ∞ Ω

where k1 (ξ) = e−2µˆg(ξ)c/kξk , i 2µc h dh c k2 (ξ) = ( − c) + g ˆ (ξ)kξk e−2µˆg(ξ)c/kξk , c kξk4 dt kξk c3 −2µˆg(ξ)c/kξk k3 (ξ) = e . kξk6 Here h(t) : R → [0, 1] is a part of the cut-off function g(x) defined in appendix A. By using Lemma 3.1 or Lemma 1.2 of appendix B, one can easily show that if [a ij ] = I, c I. Thus, we have the following corollary. an identity matrix, then [qij ] = kξk2 Corollary 3.2.

If [aij ] = I, then ˆ u, vˆ)| ˆ := A(ˆ Ω∞

Z

ˆ∞ Ω

+

ˆ v )| ˆ := F(ˆ Ω∞

+ Z

Z

Z

£ ¤ k1 (ξ) (∇ξ u ˆ)T · (∇ξ vˆ) dξ

ˆ∞ Ω

ˆ∞ Ω

ˆ∞ Ω

£ ¤ k2 (ξ)ˆ v (ξ) (∇ξ u ˆ)T · (ξ) dξ k3 (ξ)ˆ u(ξ)ˆ v (ξ)dξ,

k3 (ξ)fˆ(ξ)ˆ v (ξ)dξ,

where k1 (ξ) = e−2µˆg(ξ)c/kξk

c , kξk2

(3.4) (3.5)

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OH, YUN AND JANG

i 2µc2 h dh c c ( − c) + g ˆ (ξ)kξk e−2µˆg(ξ)c/kξk , k2 (ξ) = kξk6 dt kξk c3 −2µˆg(ξ)c/kξk k3 (ξ) = e . kξk6

Remark. Suppose Ω∞ ⊂ R2 . Then the auxiliary mapping ϕ∞ is a conformal mapping. Thus, by Lemma 1.1 of appendix B, one gets Z Z (∇ξ u ˆ)T · (∇ξ vˆ)dξ < ∞. (∇x u)T · (∇x v)dx = Ω∞

ˆ∞ Ω

It is not necessary to introduce any weight functions into our method for FE solution of −∆u = f on 2-D exterior domain, whenever f (x)|J(ϕ∞ )|−1 is square integrable on Ω∞ ⊂ R2 . (see, [2] for details). On the other hand, Corollary 3.2 shows that in 3-D case, the weight function is unavoidable to make the transformed bilinear form being finite. B. Infinite Tetrahedral Elements

Unlike 2-D cases, computing the Jacobian of blending type elemental mapping for a 3-D element with curved face ([18]) is generally much more expensive than that of a conventional elemental mapping for an element with straight faces. Thus, we avoid curved faces in our finite element mesh on 3-D domains if possible. Moreover, in order to reduce computing time, the standard blending method for elemental mappings for curved faces will not be employed even when curved faces are unavoidable (Mapped infinite elements in Fig. 3). Let us select a positive constant c and let QB be a polyhedron whose vertices are on the sphere {x ∈ R : kxk = c} and faces are triangles as shown in Fig. 2. Here c is large enough so that QB contains the simply connected bounded region B. Let ΩB (the region between ∂B and QB ) be a bounded subregion of the exterior of B. Now Ω is divided ¯B. into two parts; ΩB and Ω∞ = Ω\Ω The constant c is determined by the size of the bounded region on which an accurate numerical solution is desirable. The polyhedron QB with triangular faces can be constructed freely. However, the vertices of the polyhedron QB are on the sphere of radius c and the number of faces of QB is determined by a desired finite element mesh on a selected bounded subdomain. Suppose q1 , q2 , ..., qd are all those vertices of polyhedron QB (see, Fig 2) whose spherical coordinates are (c, φ1 , θ1 ), ..., (c, φd , θd ), where 0 ≤ φi ≤ π, 0 ≤ θi ≤ 2π. For the sake of the convenience of notations for elemental mappings and the implementation of our method, we introduce a fictitious node q∞ (which is like an extra point in one point compactification, Ω∗∞ = Ω∞ ∪ {q∞ }, of the infinite domain Ω∞ ) and assume all of the functions we consider are zero at q∞ . Then Ω∗∞ can be decomposed into tetrahedral elements each of which sits on one of the triangular faces of QB and has the fourth vertex q∞ as the common vertex. By infinite tetrahedral elements, we mean these unbounded tetrahedral elements whose union become Ω∗∞ . On the other hand, in Ω∞ , these infinite elements are determined by

ELLIPTIC PROBLEMS ON 3-D UNBOUNDED DOMAINS

9

Z

C

Y X

A

FIG. 2.

B

¯B ∩ Ω ¯ ∞. Scheme of the outer boundary Γout (QB ) of ΩB . A specific mesh on Γout = Ω

the triangular faces of QB and the rays connecting the origin and all of vertices q1 , ..., qd as shown in Fig. 3.

C. Mesh Generation

Suppose ∆B = {Ek : k = 1, 2, ..., N (∆B )} represents a specific mesh on ΩB and let ∆∞ = {Tk : k = 1, 2, ..., N (∆∞ )} be the specific mesh on Ω∞ which are infinite tetrahedral elements determined by the specific mesh on ΩB . Then ∆ = ∆B ∪ ∆∞ is a specific mesh for the unbounded domain Ω. However, neither infinite tetrahedral elements nor the fictitious node will be used in ˆ = actual computation. The conventional FEM will be applied to a specific mesh ∆ ∆B ∪ [ϕ∞ (∆∞ )] of a bounded domain ΩB ∪ [ϕ∞ (Ω∗∞ )]. Let us note that ϕ∞ (Ω∗∞ ) is virtually the unit ball (see, Fig. 1). D. Elemental Mappings for Infinite Tetrahedral elements

The construction of a finite element space of this section is similar to constructing a finite element space on the one point compactification, Ω∗∞ = Ω∞ ∪ {q∞ }, of Ω∞ by imposing a nodal constraint at q∞ . ¯B ∩ Ω ¯∞ By the inversion mapping ϕ∞ defined by (3.1), the nodes qk = (c, φk , θk ) on Ω are mapped to nodes on the unit sphere, qˆk = (1, π − φk , θk ), k = 1, ..., d, and we define ϕ∞ (q∞ ) = (0, 0, 0), the origin of the mapped unit sphere. An unbounded infinite tetrahedral element Tn∞ := (qn+1 → qn → qn−1 → q∞ ) is mapped to a bounded tetrahedral element Tˆn∞ := (ˆ qn+1 → qˆn → qˆn−1 → (0, 0, 0)), whose first face is a curved face going out from the unit ball (see, Fig. 1 & Fig. 3). Let us note that if an infinite element Tn∞ is oriented in a positive sense in Ω∗∞ , then the

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OH, YUN AND JANG

mapped bounded element Tˆn∞ is also oriented in a positive sense in the mapped bounded ˆ ∞. domain Ω For example, suppose in Fig. 2, the spherical coordinates of the points A, B, C on Ω B , respectively, are A = (c, φ1 , θ1 ), B = (c, φ2 , θ2 ), C = (c, φ3 , θ3 ).

(3.6)

∞

Let T = (A → B → C → q∞ ) be an infinite tetrahedral element sitting on the (th) triangle ABC. Let Ωst be the reference (standard) tetrahedral element in the ξ-η-ζ √ (th) (th) (th) (th) space√with√vertices V1 = (−1, 0, 0), V2 = (1, 0, 0), V3 = (0, 1/ 3, 0), V4 = (0, 1/ 3, 2 6/3) (see p243 of [27]). Then the nodal basis functions are L1 =

1 1 (1 − ξ − √ η − 2 3 √ 3 (η − L3 = 3

(th)

by

(th)

Let ϕ1 : Ωst \{V4

1 √ ζ); 6 1 √ ζ); 8

1 1 1 (1 + ξ − √ η − √ ζ) 2 3 6 r 3 ζ. L4 = 8 L2 =

√ (th) } −→ T0 = {(ξ, η, 0) : (ξ, η, 0) ∈ Ωst } × [0, 2 6/3) be defined √ 4η − 2ζ 4ξ √ , √ , ζ) ϕ1 (ξ, η, ζ) = ( 4 − 6ζ 4 − 6ζ

(3.7)

and ϕ2 : T0 −→ T ∞ be defined by

ϕ2 (ξ, η, ζ) =

3 X

(xi , yi , zi )Li (ξ, η, 0)

i=1

where, for i = 1, 2, 3, xi = ρ sin φi cos θi , yi = ρ sin φi sin θi , zi = ρ cos φi , (3.8) √ c(2 6/3) . ρ = √ 2 6/3 − ζ √ As ζ varies from 0 to 2 6/3, ρ takes all values in [0, ∞). For each radius ρ, (x1 , y1 , z1 ), (x2 , y2 , z2 ), (x3 , y3 , z3 ), defined by (3.8), are the translations of the base nodes A, B, C, defined by (3.6), by ρ − c, along the three infinite rays connecting (0, 0, 0) with A, B, and C, respectively. Then, by combining these two mappings, we define an elemental mapping Φ T ∞ : (th) Ωst −→ T ∞ by ( (th) ϕ2 ◦ ϕ1 (ξ, η, ζ) if (ξ, η, ζ) 6= V4 (3.9) ΦT ∞ (ξ, η, ζ) = (th) q∞ if (ξ, η, ζ) = V4 Let M be the vector of elemental mappings assigned to the elements in ∆ = ∆ B ∪ ∆∞ by the following rule:

• Assign the conventional elemental mappings ΦEn to the elements En ⊂ ΩB ; • Assign the singular elemental mappings ΦTn∞ defined by (3.9) to the elements Tn∞ ⊂ Ω∗∞ .

ELLIPTIC PROBLEMS ON 3-D UNBOUNDED DOMAINS

11

Z Y X

T∝

A

C B

Z Y X

A

*

C

B*

FIG. 3. (a) The Beginning Portion of Infinite Tetrahedral Element (T∞ ) sitting on the triangle ABC. (b) The Image of an Infinite Tetrahedral Element (Tˆ∞ ) under the inversion auxiliary mapping ϕ∞ .

(∗)

(th)

Suppose Ωst represents the reference tetrahedral element Ωst , the reference penta(p) (h) (∗) hedral element Ωst , or the reference hexahedral element Ωst and let Pp (Ωst ) be the (∗) space of polynomials of degree ≤ p defined on Ωst . Then the finite element space, denoted by S p (Ω, ∆, M), is the set of all functions u defined on Ω such that

• u ◦ ΦEn ∈ Pp (Ω(∗) st ) for each element En ∈ ∆B ;

(th) • u ◦ ΦTn∞ ∈ Pp (Ωst ) for each element Tn∞ ∈ ∆∞ .

Let us note that each member of S p (Ω, ∆, M) is in W 1 (Ω; µ, g) except the nodal basis function corresponding to the fictitious node q∞ . However, because of the zero-nodalconstraint at q∞ , we may claim that S p (Ω, ∆, M) ⊂ W 1 (Ω; µ, g). Since the singular elemental mappings ΦT ∞ are designed to agree with the conventional elemental mappings along the common faces on ΩB ∩Ω∞ , S p (Ω, ∆, M) is “exactly conforming” ([27]). In other words, each member of this finite element space is continuous in order to ensure good approximation properties.

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As usual, the finite element solution uf e is the projection of the exact solution into S p (Ω, ∆, M), with respect to a proper inner product. The dimension of the vector space S p (Ω, ∆, M) is called the Number of Degrees of Freedom (DOF). In the p-Version of the Finite Element Method([3],[4], [5], [27]), to obtain the desired accuracy, the mesh ∆ of the domain Ω is fixed and only the degree p of the basis polynomials is increased. E. Computation of local stiffness matrices and local load vectors for infinite tetrahedral elements

In the master element approach, the computations of local stiffness matrices and local load vectors for infinite tetrahedral elements T ∞ in Ω∗∞ can not be computed by a conventional FE code because infinite elements are not usual elements. However, we circumvent this problem by computing the transformed bilinear form ˆ ˆ ∗ given in Lemma 3.1, on the ˆ ·)| ˆ ∗ and the transformed linear functional F(·)| A(·, Ω∞ Ω∞ mapped tetrahedral elements Tˆ∞ together with the elemental mapping (th) ˆ ˆ∞ = ϕ∞ ◦ ΦT ∞ : Ωst Φ −→ T ∞ −→ Tˆ∞ , T

where ΦT∞ is defined by (3.9). In other words, the local stiffness matrices and local load vectors are computed by the following rule:

• Use A(·, ·) and F(·, ·) for the elements E in the bounded subdomain ΩB . ˆ ∗ , for the infinite tetrahedral ˆ ˆ ∗ on the elements Tˆ∞ in Ω ˆ ·)| ˆ ∗ and F(·)| • Use A(·, ∞ Ω∞ Ω∞ elements T ∞ in Ω∗∞ .

On ΩB , our method uses the conventional FEM incorporated with the standard bilinear form. Thus, stiffness matrices and load vectors for the elements E in Ω B can be computed by any existing finite element code without alteration. However, for those inˆ ·) and the finite tetrahedral elements, the transformed non-symmetric bilinear form A(·, transformed linear functional Fˆ of Lemma 3.1, are employed. Thus, our method can also be implemented on any existing finite element code. The only difference of our method from the conventional FEM is that the local stiffness matrices corresponding to infinite triangular elements T ∞ are non-symmetric. In summary, our method is to apply a conventional p-version of FEM on the bounded ˆ ∗∞ (which means that the outer boundary of Ω is fictitiously connected domain ΩB ∪ϕ∞ Ω ˆ ∗∞ by the mapping ϕ∞ ; and hence no artificial boundaries are generated) with the to ∂ Ω ˆ ·)| ˆ ∗ and the two phase linear functional two phase bilinear form A(·, ·)|ΩB ∪ϕ∞ A(·, Ω∞ ˆ ˆ∗ . F(·)|Ω ∪ϕ F(·)| B

∞

Ω∞

Now the Weighted Riesz-Galerkin (approximation) Method(WRGM) of the variational problem (2.8) is the following: Given a finite dimensional subspace S p (Ω, ∆, M), find up (x) ∈ S p (Ω, ∆, M) such that A(up , v) = F(v), for all v ∈ S p (Ω, ∆, M),

(3.10)

where A(., .) and F(.) are the bilinear form and the linear functional in (2.10) and ˆ ·)| ˆ ∗ . (2.11), respectively. Let us note that by Lemma 3.1, A(·, ·) = A(·, ·)|ΩB ∪ϕ∞ A(·, Ω∞ ˆ ˆ∗ . and F(·, ·) = F(·)|Ω ∪ϕ F(·)| B

∞

Ω∞

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13

Z

X

Y

B = R 3 - (Unbounded Domain)

FIG. 4. Diagram of the inner boundary Γin of ΩB , of Examples 1 and 2, which is the boundary of the unbounded domain Ω = R3 \B. The vertices of the octahedron ∂B are the intersections of the unit sphere with the coordinate axes.

By applying C´ea’s Lemma ([9]) and Theorem 2.1, we proved the following theorem in ([25]). Theorem 3.3.

If µ and ρ are selected so that 0 ≤ µρ
0 if t ≤ 0,

where a is a positive constant. Then the function q(t) is smooth. Let h(t) =

q(t) . (q(t) + q(b − t))

(A1)

Then h(t) is also smooth, and 0 ≤ h(t) ≤ 1,   0 if t ≤ 0 h(t) =  1 if b ≤ t. Set

g(x; a, b, c) = h(kxk − c), (x21

where a, b, c are positive constants and kxk = smooth function satisfying the following properties: (1) (2) (3)

+

(A2) x22

+

1 x23 ) 2 ,

x ∈ Ω. Then g is a

0 ≤ g(x; a, b, c) ≤ 1 for x ∈ Ω, g(x; a, b, c) = 0 if kxk ≤ c, g(x; a, b, c) = 1 if kxk ≥ c + b,

In case there is no confusion, we simply denote this cut-off function by g(x). We are interested in the constants a, b, for which the size of the gradient vector of g becomes

ELLIPTIC PROBLEMS ON 3-D UNBOUNDED DOMAINS

19

small.

B. The Change of Variables

Let S and Sˆ be subsets of Rn . Let ϕ : S → Sˆ be a bijective transformation. The Jacobian of ϕ is denoted by J(ϕ) and |J(ϕ)| denotes its determinant. ϕ will be selected so that |J(ϕ)| > 0 for all x ∈ S. ˆ through the bijective mapping ϕ, is denoted The shift of a function u : S → R onto S, −1 d by u ˆ := u ◦ ϕ . Similarly, we denote Jij (ϕ) ◦ ϕ−1 by Jij (ϕ), where Jij (ϕ) is the (i, j) component of the Jacobian of ϕ. Let x = (x1 , x2 , x3 ), x ˆ = (xˆ1 , x ˆ2 , x ˆ3 ). Then, by applying the chain rule to u(x) = u ˆ(ϕ(x)), we have (∇x u(x)) = J(ϕ) · (∇xˆ u ˆ) ◦ ϕ. Thus, the following change of variables is obtained: Z (∇x u)T · [aij (x)] · (∇x v)dx = S Z T d · [ˆ d |J(ϕ−1 )|(∇xˆ u ˆ)T J(ϕ) aij (ˆ x)] · J(ϕ)(∇ ˆ)dˆ x. x ˆv ˆ S

(A3)

(A4)

Therefore, the transformed bilinear form by the inversion mapping ϕ∞ defined by (3.1) is given in the following. T

d d∞ ) = [qij ]. Then, for u, v ∈ W 1 (Ω; µ, g), Lemma 1.1. Let |J(ϕ−1 aij (ˆ x)]·J(ϕ ∞ )|J(ϕ∞ ) ·[ˆ we have Z n e−2µg(x)kxk (∇x u(x))T · [aij (x)] · (∇x v(x)) A(u, v) := Ω∞ ³ ´o + v(x)(∇x u(x))T · [aij (x)] · ∇x e−2µg(x)kxk dx Z © ª ˆ)T · [qij ] · (∇xˆ vˆ) + vˆ(ˆ x)W (ˆ x)(∇xˆ u ˆ)T · [qij ] · (ˆ x) dˆ x e−2µˆg(ˆx)c/kˆxk (∇xˆ u = ˆ∞ Ω

ˆ u, vˆ), := A(ˆ

where W (ˆ x) =

i 2µc h dh c ( − c) + g ˆ (ˆ x )kˆ x k . c kˆ xk4 dt kˆ xk

For the proof of this key lemma, we refer ([25]). For the implementation of the mapping method, the coefficients qij of the transformed bilinear form are expanded in the following lemma. Lemma 1.2.

With the same notations as above, we obtain the following. c3 −1 −1 d (1) J(ϕ∞ ) ◦ ϕ−1 . (2) |J(ϕ∞ )| = . ∞ = J(ϕ∞ ) = [J(ϕ∞ )] kxk6

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OH, YUN AND JANG

(3) The entries of the 3 × 3 matrix [qij ] are as follows: q11 (ˆ x) =

c (ˆ a11 A2 + (ˆ a12 + a ˆ21 )AB + a ˆ22 B 2 kˆ xk6

+(ˆ a23 + a ˆ32 )BC + (ˆ a31 + a ˆ13 )AC + a ˆ33 C 2 ),

c ˆ22 BD (ˆ a11 AB + a ˆ12 AD + a ˆ21 B 2 + a kˆ xk6 +ˆ a23 BE + a ˆ32 CD + a ˆ31 BC + a ˆ13 AE + a ˆ33 CE), c (−ˆ a11 AC − a ˆ12 AE − a ˆ21 BC − a ˆ22 BE q13 (ˆ x) = kˆ xk6

q12 (ˆ x) =

+ˆ a23 BF − a ˆ32 CE − a ˆ31 C 2 + a ˆ13 AF + a ˆ33 CF ), c 2 q21 (ˆ x) = (ˆ a11 AB + a ˆ12 B + a ˆ21 AD + a ˆ22 BD kˆ xk6 +ˆ a23 CD + a ˆ32 BE + a ˆ31 AE + a ˆ13 BC + a ˆ33 CE), c q22 (ˆ x) = (ˆ a11 B 2 + (ˆ a12 + a ˆ21 )BD + a ˆ22 D2 kˆ xk6 +(ˆ a23 + a ˆ32 )DE + (ˆ a13 + a ˆ31 )BE + a ˆ33 E 2 ),

q23 (ˆ x) =

c (−ˆ a11 BC − a ˆ12 BE − a ˆ21 CD − a ˆ22 DE kˆ xk6

+ˆ a23 DF − a ˆ32 E 2 + a ˆ13 BF − a ˆ31 CE + a ˆ33 EF ),

q31 (ˆ x) =

c (−ˆ a11 AC − a ˆ12 BC − a ˆ21 AE − a ˆ22 BE kˆ xk6

−ˆ a23 CE + a ˆ32 BF + a ˆ31 AF − a ˆ13 C 2 + a ˆ33 CF ), c q32 (ˆ x) = (−ˆ a11 BC − a ˆ12 CD − a ˆ21 BE − a ˆ22 DE kˆ xk6

−ˆ a23 E 2 + a ˆ32 DF + a ˆ31 BF − a ˆ13 CE + a ˆ33 EF ), c 2 2 q33 (ˆ x) = (ˆ a11 C + (ˆ a12 + a ˆ21 )CE + a ˆ22 E kˆ xk6 −(ˆ a23 + a ˆ32 )EF − (ˆ a31 + a ˆ13 )CF + a ˆ33 F 2 ).

where A = −ˆ x21 + x ˆ22 + x ˆ23 , C = 2ˆ x1 x ˆ3 , E = 2ˆ x2 x ˆ3 ,

B = −2ˆ x1 x ˆ2 , D = x ˆ21 − x ˆ22 + x ˆ23 , 2 2 F = −(ˆ x1 + x ˆ2 − x ˆ23 ).

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