Proc. Indian Acad. Sci. (Math. Sci.) c Indian Academy of Sciences
h-p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-I: Regularity estimates and stability theorem P DUTT1 , AKHLAQ HUSAIN2,∗ , A S VASUDEVA MURTHY3 and C S UPADHYAY4 1 Department of Mathematics & Statistics, Indian Institute of Technology, Kanpur 208 016, India 2 Department of Mathematics, The LNM Institute of Information Technology, Jaipur 302 031, India 3 TIFR Centre For Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore 560 065, India 4 Department of Aerospace Engineering, Indian Institute of Technology, Kanpur 208 016, India * Corresponding author. E-mail:
[email protected];
[email protected];
[email protected];
[email protected]
MS received 13 March 2013; revised 20 April 2014 Abstract. This is the first of a series of papers devoted to the study of h-p spectral element methods for solving three dimensional elliptic boundary value problems on non-smooth domains using parallel computers. In three dimensions there are three different types of singularities namely; the vertex, the edge and the vertex-edge singularities. In addition, the solution is anisotropic in the neighbourhoods of the edges and vertex-edges. To overcome the singularities which arise in the neighbourhoods of vertices, vertex-edges and edges, we use local systems of coordinates. These local coordinates are modified versions of spherical and cylindrical coordinate systems in their respective neighbourhoods. Away from these neighbourhoods standard Cartesian coordinates are used. In each of these neighbourhoods we use a geometrical mesh which becomes finer near the corners and edges. The geometrical mesh becomes a quasi-uniform mesh in the new system of coordinates. We then derive differentiability estimates in these new set of variables and state our main stability estimate theorem using a non-conforming h-p spectral element method whose proof is given in a separate paper. Keywords. Spectral element method; non-smooth domains; geometric mesh; vertex singularity; edge singularity; vertex-edge singularity; differentiability estimates; stability estimates; exponential accuracy. Mathematics Subject Classification.
65N35, 35J25.
1. Introduction Finite element methods (FEM) are one of the most widely used techniques for solving problems in structural mechanics. There are three versions of the FEM namely; the h
P Dutt et al. version, the p version and the h-p version. The h version uses polynomials of a fixed degree and the mesh size h is reduced to increase accuracy. In the p version, a fixed mesh is used and polynomial degree p is raised to increase accuracy. The h-p version combines the approaches of the h and p versions. It simultaneously refines the mesh and increases the polynomial degree to solve problems on non-smooth domains and achieve optimal convergence. The h-p version of Spectral element method (SEM) employ global polynomials of higher degree in order to recover the so-called spectral/exponential convergence. A method for obtaining a numerical solution to exponential accuracy for elliptic problems on non-smooth domains in R2 was first proposed by Babuška and Guo [7, 8] in the frame work of the finite element method. In [18, 19], an exponentially accurate h-p spectral element method was proposed for two dimensional elliptic problems on non-smooth domains with analytic coefficients posed on curvilinear polygons with piecewise analytic boundary. The method is able to resolve the singularities which arise at the corners using a geometrical mesh as proposed by Babuška and Guo. In contrast to the two dimensional case, the character of the singularities in three dimensions is much more complex not only because of higher dimension but also due to the varied nature of the singularities which are the vertex singularity, the edge singularity and the vertex-edge singularity. Thus, we have to distinguish between the behaviour of the solution in the neighbourhoods of the vertices, edges and vertex-edges. Unlike the two dimensional case where weighted isotropic spaces are used, in three dimensions we have to utilize weighted anisotropic spaces because the solution is smooth along the edges but singular in the direction perpendicular to the edges [4]. Behaviour of the solution is even more complex at the vertices where the edges are joined together and the solution is not smooth along the edges too. Guo [24] introduced the relevant anisotropic weighted spaces to study elliptic problems on non-smooth polyhedral domains. Since then the proof that the regularity of solutions of elliptic boundary value problems on non-smooth domains is described by these sapces remained an open problem for a long time. To prove the analytic regularity for these problems, Babuška and Guo [2, 3] started the study of analytic regularity of elliptic problems on non-smooth domains in R3 in the frame work of weighted Sobolev spaces with Cauchy type control of all derivatives in the so-called countably normed spaces in the neighbourhoods of vertices, edges and vertex-edges in spherical, cylindrical and Cartesian coordinates. However, proving these regularity results is quite technical and sometimes difficult to follow as can be seen in the papers by Babuška and Guo [2–4, 24]. We remark that these regularity estimates on polyhedral domains were assumed to be true in the error analysis of h-p FEM in [24, 29] and in the error analysis of h-p SEM in [1]. Recently, Costabel and coworkers settled the proof of the analytic regularity estimates in [11] by filling the gap which was left over by Babuška and Guo. They combined a priori basic regularity results of low order for elliptic problems on polyhedral domains [14, 27] with the regularity shift estimates of Cauchy type to complete the proof using a nested open set technique and dyadic partition technique near corners [11–14]. The techniques employed in [11–13] extends to the general elliptic problems having lower order terms and variable coefficients examined in this paper (see [11] and references therein). It follows from [11, 12] that the solution to the problem under consideration in this paper belongs to an analytic class which is defined using anisotropic weighted Sobolev spaces introduced in [2, 3].
Regularity estimates and stability theorem The h-p version of the finite element method for solving three dimensional elliptic problems on non-smooth domains with exponential accuracy was proposed by Guo in [21, 24]. To overcome the singularities which arise along vertices and edges they used geometric meshes which are defined in neighbourhoods of vertices, edges and vertex-edges. We refer to [4–6, 9, 10, 22–24] for a detailed discussion of the h-p FEM and [23] on the use of auxiliary mappings for the finite element solutions of three dimensional elliptic problems on non-smooth domains. An efficient and exponentially accurate h-p spectral element method to solve general elliptic problems on non-smooth domains in three dimensions is now described. We also use a geometric mesh in the neighbourhoods of vertices, edges and vertex-edges and in each of these neighbourhoods we switch to a modified system of local coordinates using auxiliary mappings and this enables us to obtain the solution with exponential accuracy. These local coordinates are modified versions of the standard spherical and cylindrical coordinate systems in vertex and edge neighbourhoods respectively and a hybrid combination of spherical and cylindrical coordinates in vertex-edge neighbourhoods. The geometric mesh becomes geometrically fine in these neighbourhoods and in the new set of variables in these neighbourhoods the geometric mesh is mapped to a quasi uniform mesh. Hence Sobolev’s embedding theorems and the trace theorems apply for spectral element functions defined on mesh elements in the new system of variables with a uniform constant. This new system of coordinates in two dimensions was first proposed by Kondratiev [25] and we shall refer to them as modified system of coordinates. Away from the neighbourhoods of vertices, edges and vertex-edges, we retain the standard Cartesian coordinate system (x1 , x2 , x3 ) in the regular region of the polyhedron. We now seek a solution to elliptic BVP’s as in [18, 19, 30–32] which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and a fractional Sobolev norm of the residuals in the boundary conditions and enforce continuity across inter element boundaries by adding a term which measures the sum of the squares of the jump in the function and its derivatives at inter element boundaries in appropriate Sobolev norms to the functional being minimized. Here we examine the non-conforming version of the method. The case when the spectral element functions are conforming will be examined in future work. This series of papers is devoted to the study of regularity theory and implementation of the h-p spectral element method for three dimensional elliptic problems on nonsmooth domains with analytic coefficients and piecewise analytic boundary data using parallel computers. The first paper deals with the regularity of the solution in the neighbourhoods of vertices, edges and vertex-edges and the stability theorem. The second paper is more technical in nature and addresses proof of the stability theorem [16]. The numerical scheme, error estimates, construction of preconditioners on regular as well as singular regions of the polyhedron and the solution techniques are discussed in the third paper. Theoretical results have been validated by computational experiments independently in [17]. We mention here that these papers are based on the thesis work of one of the authors and we refer to [1] for complete details. The organization of this paper is as follows. In §2, we introduce the problem under consideration. We define various neighbourhoods of vertices, edges and vertex-edges and recall the function spaces in these neighbourhoods, as defined in [2], which will be needed in the sequel. In §3 we drive differentiability (regularity) estimates in various neighbourhoods. In §4, we impose a geometrical mesh on each of these neighbourhoods, define the spectral element functions on these elements and give construction of the stability
P Dutt et al. estimate (without proof) which is the main result of the paper. Section 5 gives concluding remarks. 2. Preliminaries and notations Let denote a polyhedron in R 3 , as shown in figure 1. We shall denote the boundary of by ∂. Let i , i ∈ I = {1, 2, . . . , I } be the faces of the polyhedron. Let D be a subset of I and N = I \ D. We impose Dirichlet boundary conditions on the faces i , i ∈ D and Neumann boundary conditions on the faces j , j ∈ N . Further, let ∂ = [0] ∪ [1] , [0] [1] ¯ ¯ = i∈D i and = i∈N i . We consider an elliptic boundary value problem posed on with mixed Neumann and Dirichlet boundary conditions: Lw = F in , [0] for x ∈ [0] , w = g ∂w = g [1] for x ∈ [1] , ∂n A where n denotes the outward normal and ∂w ∂n A is the usual conormal derivative. It is assumed that the differential operator Lw(x) =
3
−(aij wxj )xi +
i,j =1
3
bi wxi + cw
(2.1)
(2.2)
i=1
is a strongly elliptic differential operator which satisfies the Lax–Milgram conditions. Moreover, A = aij = aj i for all i, j and the coefficients of the differential operator are analytic. The data F , g [0] and g [1] are analytic on each open face and g [0] is continuous on i∈D ¯ i . 2.1 The neighbourhoods of vertices, edges and vertex-edges Let i , i ∈ I = {1, 2, . . . , I } be the faces (open), Sj , j ∈ J = {1, 2, . . . , J } be the edges and Ak , k ∈ K = {1, 2, . . . , K} be the vertices of the polyhedron. We shall also denote an edge by e, where e ∈ E = {S1 , S2 , . . . , SJ }, the set of edges, and a vertex by v where
Figure 1. Polyhedral domain .
Regularity estimates and stability theorem v ∈ V = {A1 , A2 , . . . , AK }, the set of vertices. Let Bρv (v) = {x : dist (x, v) < ρv }. For every vertex v, ρv is chosen so small that Bρv (v) ∩ Bρv (v ) = ∅ if the vertices v and v are distinct. Now consider a vertex v which has nv edges passing through it (figure 2). We shall let x3 axis denote one of these edges. Consider first the edge e which coincides with the x3 axis. Let φ denote the angle which x = (x1 , x2 , x3 ) makes with the x3 axis. Let Vρv ,φv (v, e) = {x ∈ : 0 < dist(x, v) < ρv , 0 < φ < φv }, where φv is a constant. Let us choose φv sufficiently small so that Vρv ,φv (v, e ) Vρv ,φv (v, e ) = ∅, where e and e are distinct edges which have v as a common vertex. Now we define v , the vertex neighbourhood of the vertex v. Let E v denote the subset of E, the set of edges, such that E v = {e ∈ E : v is a vertex of e}. Then = Bρv (v) v
Vρv ,φv (v, e)
.
e∈E v
Here, ρv and φv are chosen so that ρv sin(φv ) = Z, a constant for all v ∈ V , the set of vertices. Let
ρ = x1 2 + x2 2 + x3 2 , φ = cos−1 (x3 /ρ), θ = tan−1 (x2 /x1 ),
Figure 2. Vertex neighbourhood v .
P Dutt et al. denote the usual spherical coordinates in v . We now introduce a set of modified coordinates in the vertex neighbourhood v by x1v = φ, x2v = θ, x3v = χ = ln ρ .
(2.3)
Let e denote an edge, which for convenience we assume to coincide with the x3 axis as before, whose end points are the vertices v and v as shown in figure 3. Assume that the vertex v coincides with the origin. Let the length of the edge e be le , δv = ρv cos(φv ) and δv = ρv cos(φv ). Let (r, θ, x3 ) denote the usual cylindrical coordinates
r = x1 2 + x2 2 , θ = tan−1 (x2 /x1 ) and e the edge neighbourhood (figure 3) e = {x ∈ : δv < x3 < le − δv , 0 < r < Z} . We introduce a set of modified system of coordinates in the edge neighbourhood e by x1e = τ = ln r, x2e = θ, x3e = x3 .
Figure 3. Edge neighbourhood e .
(2.4)
Regularity estimates and stability theorem
Figure 4. Vertex-edge neighbourhood v−e .
Next, we define the vertex-edge neighbourhood v−e , shown in figure 4, as follows v−e = {x ∈ : 0 < φ < φv , 0 < x3 < δv = ρv cos φv } . We thus obtain a set of vertex-edge neighbourhoods v−e , where v − e ∈ V − E, the set of vertex-edge neighbourhoods. Let us introduce a set of modified coordinates in the vertex-edge neighbourhood v−e , x1v−e = ψ = ln(tan φ), x2v−e = θ, x3v−e = ζ = ln x3 .
(2.5)
2.2 Function spaces and dynamical weights associated with various neighbourhoods We need to review a set of function spaces and weight functions as described in [2, 21]. By H m (), we denote the usual Sobolev space of integer order m ≥ 0 furnished with the norm ||D α w||2L2 () , ||w||2H m () = |α|≤m
α
where α = (α1 , α2 , α3 ), |α| = α1 + α2 + α3 , D α w = Dxα11 Dxα22 Dx33 w = wx α1 x α2 x α3 1
2
3
is the distributional (weak) derivative of w. As usual H 0 () = L2 (), H01 () =
P Dutt et al. {w ∈ L2 () : Dw ∈ L2 (), w = 0 on ∂}. A seminorm on H m () is given by ||D α w||2L2 () . |w|2H m () = |α|=m
Let ρ = ρ(x) = dist(x, v) for x ∈ v and βv ∈ (0, 1/2). We introduce a weight function β +|α|−2 , for |α| ≥ 2 ρ v (x) = α,2 βv 1, for |α| < 2 in the neighbourhood v as in (2.3) of [2]. Let α,2 α v 2 2 w ( ) = w| = D w < ∞ Hk,2 k,2 βv βv v L2 (v ) Hβv ( )
|α|≤k
and α,2 α v α B2βv (v ) = {w|w ∈ Hk,2 βv ( ) for all k ≥ 2 and βv D w L2 (v ) ≤ C d α!}
respectively, denote the weighted Sobolev space and countably normed space defined on v as in [2]. Let r = r(x) = dist(x, e) for x ∈ e and βe ∈ (0, 1). Define a weight function β +|α |−2 , for |α | = α1 + α2 ≥ 2 r e α,2 βe (x) = 1, for |α| < 2 in the neighbourhood e as in (2.1) of [2]. Let α,2 α e 2 2 w Hk,2 ( ) = w| = D w < ∞ k,2 2 e βe βe e L ( ) Hβe ( )
|α|≤k
and α,2 α e α B2βe (e ) = {w|w ∈ Hk,2 βe ( ) for all k ≥ 2 and βe D w L2 (e ) ≤ C d α!}
respectively, denote the weighted Sobolev space and countably normed space defined on e as in [2]. Let ρ = ρ(x) and φ = φ(x) for x ∈ v−e and βv−e = (βv , βe ), βv ∈ (0, 1/2), βe ∈ (0, 1). We introduce a weight function ⎧ ⎨ ρ βv +|α|−2 (sin(φ))βe +|α |−2 , for |α | = α1 + α2 ≥ 2 α,2 β +|α|−2 βv−e (x) = ρ v , for |α | < 2 ≤ |α| ⎩ 1, for |α| < 2 in the neighbourhood v−e as in (2.2) of [2]. Let α,2 v−e 2 α 2 w Hk,2 ( ) = w| = D w < ∞ k,2 βv−e βv−e v−e L2 (v−e ) Hβ
v−e
(
)
|α|≤k
Regularity estimates and stability theorem and B2βv−e (v−e ) =
v−e w| w ∈ Hk,2 ) for all k ≥ 2 and βe ( α α α,2 βv−e D w L2 (v−e ) ≤ C d α!
respectively, denote the weighted Sobolev space and countably normed space defined on v−e as in [2]. Let us recall that by C2βv (v ) we denote a countably normed space as described in [2, ¯ v ) such that for all α, |α| ≥ 0, 21] which is the set of functions w(x) ∈ C0 ( |Dxα (w(x) − w(v))| ≤ C d α α! ρ −(βv +|α|−1/2) (x).
(2.6)
¯ v denotes the closure of v and w(v) denotes the value of w at the vertex v. Here, Next, we recall that by C2βe (e ), βe ∈ (0, 1) is denoted a countably normed space as ¯ e ) such that for |α| ≥ 0, described in [2, 21] which is the set of functions w ∈ C0 ( r βe +α1 +α2 −1 Dxα (w(x) − w(0, 0, x3 )) C0 (¯ e ) ≤ C d α α! and for k ≥ 0 dk w(0, 0, x ) 3 (dx )k 3
¯ e ∩{x: x C 0 (
1 =x2 =0})
(2.7)
≤ C d k k!,
(2.8)
¯ e denotes the closure of e . where Finally, by C2βv−e (v−e ) is denoted a countably normed space which is the set of ¯ v−e ) such that functions w(x) ∈ C0 ( ρ βv +|α|−1/2 (sin φ)βe +α1 +α2 −1 Dxα (w(x)−w(0, 0, x3 )) C0 (¯ v−e ) ≤ C d α α! (2.9) and
k d |x3 |βv +k−1/2 k (w(0, 0, x3 ) − w(v)) dx3
≤ C d k k! C0
(
¯ v−e ∩{x:x
1 =x2 =0}
) (2.10)
¯ v−e
v−e .
as described in [2, 21]. Here denotes the closure of Unless otherwise stated, as in Babuška and Guo [2, 3], we let w(x v ), w(x v−e ), w(x e ) denote w(x(x v )), w(x(x v−e )), w(x(x e )) respectively. The same notation is used for the spectral element functions u(x v ), u(x v−e ), u(x e ) etc. in the ensuing sections.
3. Differentiability estimates in modified coordinates 3.1 Differentiability estimates in modified coordinates in vertex neighbourhoods ˜ v denote the image of Let wv = w(v), denote the value of w at the vertex v and let v v in x coordinates. We can now state the differentiability estimates in these modified coordinates in vertex neighbourhoods. The proof is based on the regularity results proved by Babuška and Guo in [2]. These estimates are obtained when the differential operator
P Dutt et al. is the Laplacian. However they are valid for the more general situation examined in this paper. PROPOSITION 3.1 There exists a constant βv ∈ (0, 1/2) such that for all 0 < ν ≤ ρv the estimate v ex3 |Dxαv (w(x v ) − wv )|2 dx v ≤ C(d m m!)2 ν (1−2βv ) ˜ v ∩{x v : x v ≤ ln(ν)} |α|≤m 3
(3.1) holds for all integers m ≥ 1. Here, C and d denote positive constants and dx v denotes a volume element in x v coordinates. 0 ¯v v Proof. By Theorem 3.19 of [2], for βv ∈ (0, 1/2), H2,2 βv ( ) is embedded in C ( ) and
w C0 (¯ v ) ≤ C w H2,2 (v ) . βv
¯ v the closure of v . Hence, we can define Here, C denotes the positive constant and wv = w(v), the value of w at the vertex v and |wv | ≤ C w H2,2 (v ) . βv
Let
Dα w
= wφ α1 θ α2 ρ α3 . Here, α = (α1 , α2 , α3 ) and α = (α1 , α2 ). Let α,2 −|α | α v 2 2 w ( ) = w| = ρ D w < ∞ Hβk,2 k,2 βv ˜ v) v v L2 ( Hβv ( )
|α|≤k
and
−|α | α Bβ2v (v ) = {w|w ∈ Hβk,2 (v ) for all k ≥ 2, α,2 D w L2 (˜ v ) ≤ Cd α α!}. βv ρ v
Then from Theorem 4.13 of [2], we have that w ∈ Bβ2v (v ) if and only if w ∈ B2βv (v ). Hence, |ρ βv −2 ρ α3 wφ α1 θ α2 ρ α3 |2 ρ 2 sin φ dρ dφ dθ ≤ (Cd m m!)2 . (3.2) 2≤|α|≤m ˜ v
Define χ = ln ρ . Then ∂ ∂ =ρ ∂χ ∂ρ
and
dρ = dχ . ρ
Now using (3.2) it can be shown that 2 e(2βv −1)χ wφ α1 θ α2 χ α3 dχ dφ dθ ≤ (Cd m m!)2 . 2≤|α|≤m ˜ v
Regularity estimates and stability theorem Here, C and d denote generic constants. Hence, |Dxαv w|2 dx v ≤ (Cd m m!)2 ν 1−2βv .
(3.3)
2≤|α|≤m ˜ v ∩{x v : χ ≤ln ν} v We now obtain estimates for 0 ≤ |α| ≤ 1. By Lemma 5.5 of [2], since w ∈ H2,2 βv ( ), the estimate ρ 2(βv −2) |w − w(v)|2 dx ≤ C u 2 2,2 v Hβv ( )
v
holds. Hence,
e(2βv −1)χ |w − wv |2 dχ dφdθ ≤ (Cd m m!)2 .
˜v
Thus, we conclude that |w − wv |2 dx v ≤ C(d m m!)2 ν 1−2βv .
(3.4)
˜ v ∩{x v : χ ≤ln ν}
We know by Theorem 5.6 of [2] that B2βv (v ) ⊆ C2βv (v ). Now, ρ∇x w = Qv ∇xv w,
where Qv = O vP v .
(3.5)
Here, O v is the orthogonal matrix ⎡ ⎤ cos φ cos θ − sin θ sin φ cos θ O v = ⎣ cos φ sin θ cos θ sin φ sin θ ⎦ − sin φ 0 cos φ and
⎡
1 Pv = ⎣ 0 0
0 1 sin φ
0
⎤ 0 0 ⎦. 1
Now, in v φv < φ < π − φv . Hence, from (2.6) and (3.5) we can conclude that |∇x v w| ≤ Cρ −βv +1/2 .
(3.6)
Using (3.6), the estimate
|Dxαv w|2 dx v ˜ v ∩{x v : χ ≤ln ν}|α|=1
≤C
ln ν
−∞
e(−2βv +1)χ dχ ≤ Cν 1−2βv
follows. Combining (3.3), (3.4) and (3.7) we obtain the result.
(3.7)
P Dutt et al. 3.2 Differentiability estimates in modified coordinates in edge neighbourhoods ˜ e denote the image of e in x e coordinates. The differentiability estimates for the Let solution w in edge neighbourhoods in these modified coordinates can now be stated. PROPOSITION 3.2 Let s(x3 ) = w (x1 , x2 , x3 ) |(x1 =0,x2 =0) . Then le −δ v |Dxk e s(x3e )|2 dx3e ≤ C(d m m!)2 δv
k≤m
(3.8)
3
for all integers m ≥ 1. Moreover, there exists a constant βe ∈ (0, 1) such that for μ ≤ Z, |Dxαe (w(x e ) − s(x3e ))|2 dx e ≤ C(d m m!)2 μ2(1−βe ) (3.9) ˜ e ∩{x e : x e 0 is a degree factor. e be one of the faces of ˜ em such that μ(˜ e ) < ∞, where μ denotes measure. Let ˜ m,i m,i We define a norm ||| u |||2˜ e as follows: m,i
Let
Gem,i
= sup
(eτ ).
e x e ∈˜ m,i
e which is needed in (4.25). We also define Hm,i
e = {x e : α < x e < α , β < x e < β , x e = γ }, then define H e = Ge (1) If ˜ m,i 0 1 0 1 0 m,i m,i 1 2 3 and β1 α1 u2 (τ, θ, γ0 ) dτ dθ ||| u |||2˜ e = Gem,i m,i
β0
β1
+
β0
+
α1 α0
α0
(u(τ, θ, γ0 ) − u(τ , θ, γ0 ))2 dτ dτ (τ − τ )2 α0 α0 β1 β1 (u(τ, θ, γ0 )−u(τ, θ ,γ0 ))2 dτ dθ dθ . (θ − θ )2 β0 β0 (4.21a)
dθ
α1
α1
P Dutt et al. e = {x e : x e = α , β < x e < β , γ < x e < γ }, then define H e = 1 and (2) If ˜ m,i 0 0 1 0 1 m,i 1 2 3 γ1 β1 ||| u |||2˜ e = u2 (α0 , θ, x3 ) dθ dx3 m,i
γ0
β0
γ1
+
β1 β1
dx3 γ0
+Gem,i
β0
dθ
β1
β0
β0
u(α0 , θ, x3 ) − u(α0 , θ , x3 ) (θ − θ )2
2 dθ dθ
γ1 γ1 (u(α ,θ,x )−u(α ,θ,x ))2 0 3 0 3 dx3 )2 (x − x 3 γ0 γ0 3
dx3
.
(4.21b) e = {x e : α < x e < α , x e = β , γ < x e < γ }, then define H e = 1 and (3) If ˜ m,i 0 1 0 0 1 m,i 1 2 3
γ1 α1 ||| u |||2˜ e = u2 (τ, β0 , x3 ) dτ dx3 m,i γ0 α0 γ1 α1 α1 (u(τ, β0 , x3 ) − u(τ , β0 , x3 ))2 + dx3 dτ dτ (τ − τ )2 γ0 α0 α0 γ1 γ1 α1 (u(τ, β0 , x3 ) − u(τ, β0 , x3 ))2 . +Gem,i dτ dx dx 3 3 (x3 − x3 )2 α0 γ0 γ0 (4.21c) Let Le be a differential operator such that |Le u(x e )|2 dx e = r 2 |Lu(x)|2 dx. ˜ em
(4.22)
em
Here, dx e denotes a volume element in x e coordinates and dx a volume element in x coordinates. In Chapter 3 of [1], it is shown that 3 Le u(x e ) = −divx e Ae ∇x e u + bˆie uxie + cˆe u,
(4.23)
i=1 e be one of the sides of ˜ em and where Ae is a symmetric, positive definite matrix. Let ˜ m,i e e e P˜ a point belonging to ˜ m,i . Let ν be the normal to ˜ m,i at P˜ . Then
∂u ∂ν e
(P˜ ) = (ν e )T Ae ∇x e u(P˜ ) .
(4.24)
Ae
We now define the quadratic form VeN,W ({Fu })
Ne
=
˜ e )
