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ADDITIVE SCHWARZ PRECONDITIONING FOR P-VERSION TRIANGULAR AND TETRAHEDRAL FINITE ELEMENTS ¨ JOACHIM SCHOBERL, JENS M. MELENK, CLEMENS G. A. PECHSTEIN, AND SABINE C. ZAGLMAYR

Abstract. This paper analyzes two-level Schwarz methods for matrices arising from the p-version finite element method on triangular and tetrahedral meshes. The coarse level consists of the lowest order finite element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a model problem.

1. Introduction High order finite element methods can lead to very high accuracy and are thus attracting increasing attention in many fields of computational science and engineering. The monographs [SB91, BS94, Sch98, KS99, SDR04] give a broad overview of theoretical and practical aspects of high order methods. As the problem size increases (due to small mesh-size h and high polynomial order p), the solution of the arising linear system of equations becomes more and more the time-dominating part. Here, iterative solvers can reduce the total simulation time. We consider preconditioners based on domain decomposition methods [DW90, GO95, SBG96, TW04, Qua99]. The concept is to consider each high order element as an individual sub-domain. Such methods were studied in [Man90, BCM91, Pav94, Ain96a, Ain96b, Cas97, Bic97, GC98, SC01, Mel02, EM04]. We assume that the local problems can be solved directly. On tensor product elements, one can apply optimal preconditioners for the local subproblems as in [KJ99, BSS04, BS04]. In the current work, we study overlapping Schwarz preconditioners with large or small overlap. The condition numbers are bounded uniformly in the mesh size h and the polynomial order p. To our knowledge, this is a new result for tetrahedral meshes. We construct explicitly the decomposition of a global function into a coarse grid part and local contributions associated with the vertices, edges, 1991 Mathematics Subject Classification. 65N30. Key words and phrases. High Order Finite Element Method, Preconditioning. The first and the last author acknowledge their support by the Austrian Science Foundation FWF within project grant Start Y-192, “hpFEM : Fast Solvers and Adaptivity”. 1

2

¨ J. SCHOBERL, J. MELENK, C. PECHSTEIN, AND S. ZAGLMAYR

faces, and elements of the mesh. The idea of the construction was presented in [SMPZ05]. The rest of the paper is organized as follows: In Section 2 we state the problem and formulate the main results. We prove the 2D case in Section 3 and extend the proof for 3D in Section 4. Finally, in Section 5 we give numerical results for several versions of the analyzed preconditioners. 2. Definitions and Main Result We consider the Poisson equation on the polyhedral domain Ω with homogeneous Dirichlet boundary conditions on ΓD ⊂ ∂Ω, and Neumann boundary conditions on the remaining part ΓN . With the sub-space V := {v ∈ H 1 (Ω) : v = 0 on ΓD }, the bilinear-form A(·, ·) : V × V → R and the linear-form f (·) : V → R defined as Z Z ∇u · ∇v dx

A(u, v) =

f (v) =

Ω

f v dx, Ω

the weak formulation reads (1)

find u ∈ V such that

A(u, v) = f (v)

∀ v ∈ V.

We assume that the domain Ω is sub-divided into straight-sided triangular or tetrahedral elements. In general, constants in the estimates depend on the shape of the elements, but they do not depend on the local mesh-size. We define the set of vertices the set of edges the set of faces (3D only) the set of elements

V = {V }, E = {E}, F = {F }, T = {T }.

We define the sets Vf , Ef , Ff of free vertices, edges, and faces not completely contained in the Dirichlet boundary. The high order finite element space is Vp = {v ∈ V : v|T ∈ P p ∀ T ∈ T }, where P p is the space of polynomials up to total order p. As usual, we choose a basis consisting of lowest order affine-linear functions associated with the vertices, and of edge-based, face-based, and cell-based bubble functions. The Galerkin projection onto Vp leads to a large system of linear equations, which shall be solved with the preconditioned conjugate gradient iteration. This paper is concerned with the analysis of additive Schwarz preconditioning. The basic method is defined by the following space splitting. In Section 5 we will consider several cheaper versions resulting from our analysis. The coarse sub-space is the global lowest order space V0 := {v ∈ V : v|T ∈ P 1 ∀ T ∈ T }.

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For each inner vertex we define the vertex patch [ ωV = T T ∈T :V ∈T

and the vertex sub-space VV = {v ∈ Vp : v = 0 in Ω \ ωV }. For vertices V not on the Neumann boundary, this definition coincides to Vp ∩ H01 (ωV ). The additive Schwarz preconditioning operator is C −1 : Vp∗ → Vp defined by X C −1 d = w0 + wV V ∈V

with w0 ∈ V0 such that A(w0 , v) = hd, vi

∀ v ∈ V0 ,

and wV ∈ VV defined such that A(wV , v) = hd, vi

∀ v ∈ VV .

This method is very simple to implement for the p-version method using a hierarchical basis. The low-order block requires the inversion of the sub-matrix according to the vertex basis functions. The high order blocks are block-Jacobi steps, where the blocks contain all vertex, edge, face, and cell unknowns associated with mesh entities containing the vertex V . The rate of convergence of the cg iteration can be bounded by means of the spectral bounds for the quadratic forms associated with the system matrix and the preconditioning matrix. The main result of this paper is to prove optimal results for the spectral bounds: Theorem 1. The constants λ1 and λ2 of the spectral bounds λ1 hCu, ui ≤ A(u, u) ≤ λ2 hCu, ui

∀ u ∈ Vp

are independent of the mesh-size h and the polynomial order p. The proof is based on the additive Schwarz theory, which allows to express the C-form by means of the space decomposition: X hCu, ui = inf ku0 k2A + kuV k2A . P u=u0 + V uV u0 ∈V0 ,uV ∈VV

The constant λ2 follows immediately from a finite number of overlapping subspaces. In the core part of this paper, we construct an explicit and stable decomposition of u into sub-space functions. Section 3 introduces the decomposition for the case of triangles, in Section 4 we prove the results for tetrahedra.


4

3. Sub-space splitting for triangles In this section, we give the proof of Theorem 1 for triangles. The case of tetrahedra is postponed to Section 4. The strategy of the proof is the following: First, we subtract a coarse grid function to eliminate the h-dependency. By stepwise elimination, the remaining function is then split into sums of vertex-based, edge-based and inner functions. For each partial sum, we give the stability estimate. This stronger result contains Theorem 1, since we can choose corresponding vertices for the edge and inner contributions (see also Section 5). 3.1. Coarse grid contribution. In the first step, we subtract a coarse grid function: Lemma 2. For any u ∈ Vp there exists a decomposition (2)

u = u0 + u1

such that u0 ∈ V0 and ku0 k2A + k∇u1 k2L2 + kh−1 u1 k2L2 kuk2A . Proof. We choose u0 = Πh u, where Πh is the Clément-operator [Cle75]. The norm bounds are exactly the continuity and approximation properties of this operator. From now on, u1 denotes the second term in the decomposition (2). 3.2. Vertex contributions. In the second step, we subtract functions uV to eliminate vertex values. Since vertex interpolation is not bounded in H 1 , we cannot use it. Thus, we construct a new averaging operator mapping into a larger space. In the following, let V be a vertex not on the Dirichlet boundary ΓD , and let ϕV be the piece-wise linear basis function associated with this vertex. Furthermore, for s ∈ [0, 1] we define the level sets γV (s) := {y ∈ ωV : ϕV (y) = s}, and write γV (x) := γV (ϕV (x)) for x ∈ ωV . For internal vertices V, the level set γV (0) coincides with the boundary ∂ωV (cf. Figure 1). The space of functions being constant on these sets reads SV := {w ∈ L2 (ωV ) : w|γV (s) = const, s ∈ [0, 1] a.e.}; its finite dimensional counterpart is SV,p := SV ∩ Vp = span{1, ϕV , ..., ϕpV }. We introduce the spider averaging operator Z 1 V v(y) dy, Π v (x) := |γV (x)| γV (x)

for v ∈ L2 (ωV ).

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5

γ (x)

V

Π (u)

V x

(1−ϕ ) ΠV(u) V x

Figure 1. The level sets γV (x)

Figure 2. Construction of ΠV0

To satisfy homogeneous boundary conditions, we add a correction term as follows (see Figure 2) ΠV0 v (x) := ΠV v (x) − (ΠV v)|γV (0) (1 − ϕV (x)). Lemma 3. The averaging operators fulfill the following algebraic properties (i) ΠV Vp = SV,p , (ii) ΠV0 Vp = SV,p ∩ VV , (iii) if u is continuous at V , then (ΠV u)(V ) = ΠV0 u(V ) = u(V ). The proof follows immediately from the definitions. We denote the distance to the vertex V , and the minimal distance to any vertex in V by rV (x) := |x − V | and rV (x) := min rV (x). V ∈V

Lemma 4. The averaging operators satisfy the following norm estimates (i) kΠV ukL2 (ωV ) kukL2 (ωV ) (ii) k∇ ΠV ukL2 (ωV ) k∇ukL2 (ωV ) (iii) krV−1 {u − ΠV u}kL2 (ωV ) k∇ukL2 (ωV ) (iv) k∇{ϕV u − ΠV0 u}kL2 (ωV ) k∇ukL2 (ωV ) (v) krV−1 {ϕV u − ΠV0 u}kL2 (ωV ) k∇ukL2 (ωV )

γ (0)


6

Proof. We parameterize the patch ωV by FV : γV (0) × [0, 1] → ωV : (y, s) 7→ y + s(V − y). Splitting the patch into elements, and applying element-wise transformation rules, one proves Z 1Z Z f (FV (y, s)) (1 − s) dy ds, f (x) dx ' hV 0

ωV

γV (0)

where hV := diam{ωV }. (i) Using γV (FV (y, s)) = γV (s) together with standard inequalities we derive kΠ

V

uk2L2 (ωV )

Z

1

Z

1

Z

1

Z

1

Z

1

Z

' hV 0

Z

γV (0)

= hV 0

Z

γV (0)

≤ hV 0

Z

γV (0)

= hV 0

Z

V (Π u)(FV (y, s)) 2 (1 − s) dy ds

γV (0)

Z 2 1 u(x) dx (1 − s) dy ds |γV (s)| γV (s) Z 1 u2 (x) dx (1 − s) dy ds |γV (s)| γV (s) Z 1 u2 (FV (x, s)) dx (1 − s) dy ds |γV (0)| γV (0)

= hV u2 (FV (x, s)) dx (1 − s) ds 0 γV (0) Z ' u2 (x) dx. ωV

(ii) To verify the estimate for the H 1 -semi-norm, we rewrite the point-wise gradient: Z 1 V u(y) dy (∇Π u)(x) = ∇ |γV (x)| γV (x) Z 1 = ∇ u(FV (y, ϕV (x)) dy |γV (0)| γV (0) Z 1 d (u ◦ FV ) = (y, ϕV (x)) ∇ϕV (x) dy |γV (0)| |γV (0)| ds Z 1 = (∇u) FV (y, ϕV (x)) · (V − y) ∇ϕV (x) dy. |γV (0)| |γV (0)|

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Forming the absolute values allows us to estimate Z 1 V (∇u)(FV (y, ϕV (x))) |V − y||∇ϕV | dx |∇Π u|(x) ≤ |γV (0)| γV (0) Z 1 (∇u)(FV (y, ϕV (x))) hV h−1 dy V |γV (0)| γV (0) Z 1 |(∇u)(y)| dy = |γV (x)| γV (x) = (ΠV |∇u|) (x). The rest follows from the L2 -estimate (i) applied to |∇u|. (iii) On the manifold γV (0), there holds the Poincaré inequality Z Z Z 2 1 2 u(y) dy dx hV |∇u|2 dx. u(x) − |γV (0)| γV (0) γV (0) γV (0) Using rV (x) ' (1 − ϕV (x))hV we derive Z 2 1 u − ΠV u dx 2 ωV rV Z Z 1Z 2 1 1 ' hV u(F (y, s)) − u(F (x, s)) dx (1 − s) dy ds V V 2 |γV (0)| γV (0) 0 γV (0) rV Z 1Z 2 h2V hV ∇y u(FV (y, s)) (1 − s) dy ds 2 0 γV (0) rV Z 1Z 1 ∂FV 2 ' hV (∇u)(FV (y, s)) (1 − s) dy ds 2 ∂y 0 γV (0) (1 − s) Z 1Z (∇u)(FV (y, s)) 2 (1 − s) dy ds = hV 0

'

γV (0)

k∇uk2L2 (ωV ) .

(iv) Since ϕV 1 = ΠV0 1, we can subtract the mean value u :=

1 |ωV |

R ωV

u(x) dx:

k∇{ϕV u − ΠV0 u}k = k∇{ϕV (u − u) − ΠV0 (u − u¯)}k ≤ k∇ ϕV (u − u¯) k + k∇ΠV (u − u¯) − ΠV (u − u¯)|γV (0) ∇(1 − ϕV )k V k(∇ϕV )(u − u¯)k + kϕV ∇uk + k∇uk + Π (u − u¯)|γV (0) k∇ϕV k h−1 ku − u¯k + k∇uk k∇uk.

8


We have used (ii) and the trace inequality for Z 1 V ≤ u − u ¯ dx Π (u − u¯)|γV (0) = |γV (0)| γV (0) (3) ≤ |γV (0)|−1/2 ku − u¯kL2 (γV (0)) k∇(u − u¯)k + h−1 ku − u¯k. (v) We finally prove krV−1 {ϕV u − ΠV0 u}kL2 (ωV ) k∇ukL2 (ωV ) . From the definition of rV , we get

1

{ϕV u − ΠV0 u} ' 1 {ϕV u − ΠV0 u} + rV rV

X V 0 ∈ωV \{V }

1

{ϕV u − ΠV0 u} . rV 0

We bound the first term as follows:

1

{ϕV u − ΠV0 u} L2 (ωV ) rV

1 = (1 − (1 − ϕV ))u − ΠV u + (1 − ϕV )(ΠV u)|γV (0) rV

1 = (u − ΠV u) − (1 − ϕV )(u − u¯) + (1 − ϕV ) (ΠV u)|γV (0) − u¯ rV

1

1 − ϕV

1 − ϕV (u − ΠV u) + (u − u¯) + (ΠV u)|γV (0) − u¯ rV rV r V k∇uk + h−1 ku − u¯k + h−1 (ΠV u)|γV (0) − u¯ |ωV |1/2 k∇ukL2 (ωV ) We have used that (1 − ϕV )/rV ' h−1 , and applied the Poincaré inequality on ωV , and once again (3). Before treating the second term, we prove the following estimate on a triangle T: Z 1 v 2 dx k∇vk2L2 (T ) , (4) (rV 0 )2 T

for functions v vanishing on an edge E containing the vertex V 0 . We transform to a reference triangle Tˆ = {(x1 , x2 ) : 0 ≤ x2 ≤ x1 ≤ 1} and use Friedrichs’ inequality: Z

1 (rV

0 )2

v 2 dx ' h2

Z1 Zx1 0

T

Z1 Zx1 0

0

1 v 2 (x1 , x2 ) dx2 dx1 x21

h2 0

∂v 2 dx2 dx1 k∇vk2L2 (T ) ∂x2

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9

Since the function v := ϕV u − ΠV0 u vanishes on the boundary ∂ωV , inequality (4) can be applied on each triangle T ⊂ ωV :

1

∇{ϕV u − ΠV0 u}

{ϕV u − ΠV0 u} L2 (ωV ) L2 (ωV ) rV 0 Using (iv) and summing over V 0 , we get the desired estimate. Due to shape regularity this sum is finite. This finishes the proof of Lemma 4 The global spider vertex operator is X

ΠV :=

ΠV0 .

V ∈Vf

Obviously, u−ΠV u vanishes in any vertex V ∈ Vf . These well-defined zero vertex values are reflected by the following norm definition: 1 (5) ||| · |||2 := k∇ · k2L2 (Ω) + k · k2L2 (Ω) rV Theorem 5. Let u1 be as in Lemma 2. Then, the decomposition X ΠV0 u1 + u2 (6) u1 = V ∈Vf

is stable in the sense of X

(7)

kΠV0 u1 k2A + |||u2 |||2 kuk2A .

V ∈Vf

Proof. The vertex terms in equation (7) are bounded by kΠV0 u1 k2A = kΠV u1 − (ΠV u1 )|γV (0) (1 − ϕV )k2A 2 ≤ k∇ΠV u1 k2L (ω ) + (ΠV u1 )|γ (0) k1 − ϕV k2A 2

k∇u1 k2L2 (ωV )

V

V

+

2 h−2 V ku1 kL2 (ωV ) .

We have used that k1 − ϕV kA ' 1. Summing up all terms, one obtains X kΠV0 u1 k2A k∇u1 k2L2 (Ω) + kh−1 u1 k2L2 (Ω) kuk2A . V ∈Vf

To bound the second term, we compare with the partition of unity provided by the hat functions: 2 X 2 X X V V Π0 u = ϕV u + (ϕV u − Π0 u) u − V ∈Vf

V ∈VD

X V ∈VD

V ∈Vf

|||ϕV u|||2 +

X V ∈Vf

|||(ϕV u − ΠV0 u)|||2

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The functions ϕV u have a zero-edge for each vertex, and thus, an argument similar to that of Lemma 4, part (v) applies and leads to |||ϕV u|||2 k∇(ϕV u)k2L2 (ωV ) ≤ k∇uk2L2 (ωV ) . For the rest of this section, u2 denotes the second term in the decomposition (6). 3.3. Edge contributions. As seen in the last subsection, the remaining function u2 vanishes in all vertices. We now introduce an edge-based interpolation operator to carry the decomposition further, such that the remaining function, u3 , contributes only to the inner basis functions of each element. Therefore we need a lifting operator which extends edge functions to the whole triangle preserving the polynomial order. Such operators were introduced in Babuˇska et al. [BCM91], and later simplified and extended for 3D by Mu˜ nozSola [Mun97]. The lifting on the reference element T R with vertices (−1, 0), (1, 0), (0, 1) and edges E1R := (−1, 1) × {0}, E2R , E3R reads: Z x1 +x2 1 w(s)ds, (R1 w)(x1 , x2 ) := 2x2 x1 −x2 for w ∈ L1 ([−1, 1]). The modification by Mu˜ noz-Sola preserving zero boundary values on the edges E2R and E3R is w (x1 , x2 ). (Rw)(x1 , x2 ) := (1 − x1 − x2 ) (1 + x1 − x2 ) R1 1 − x21 For an arbitrary triangle T = FT (T R ) containing the edge E = FT (E1R ), its transformed version reads RT w := R w ◦ FT ◦ FT−1 . 1/2

The Sobolev space H00 (E) on an edge E = [VE,1 , VE,2 ] is defined by its corresponding norm Z 1 2 2 2 w ds, kwkH 1/2 (E) := kwkH 1/2 (E) + rVE 00 E

with rVE := min{rVE,1 , rVE,2 }. Lemma 6. The Mu˜ noz-Sola lifting operator RT satisfies: (i) it maps polynomials w ∈ P0p (E) := {v ∈ P p (E) : v = 0 in VE,1 , VE,2 } into {v ∈ P p (T ) : v = 0 on ∂T \ E}. (ii) it is bounded in the sense kRT wkH 1 (T ) kwkH 1/2 (E) . 00

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11

The proof follows from [BCM91] and [Mun97]. We call ωE := ωVE,1 ∩ ωVE,2 the edge patch. We define an edge-based interpolation operator as follows: 1 ΠE 0 : {v ∈ Vp : v = 0 in V} → H0 (ωE ) ∩ Vp ,

(8)

(ΠE 0 u)|T := RT trE u.

Lemma 7. The edge-based interpolation operator ΠE 0 defined in (8) is bounded in the ||| · |||-norm: k∇ΠE 0 ukL2 (ωE ) |||u|||ωE Proof. Foremost, we apply Lemma 6 on a single triangle T ⊂ ωE : 2 2 k∇ΠE 0 ukL2 (T ) = k∇RT trE ukL2 (T ) Z 2 k trE ukH 1/2 (E) +

E

1 rVE

(trE u)2 ds.

For the first term, the trace theorem can be immediately applied. The second term, the weighted L2 -norm on the edge, can be bounded by a weighted norm on the triangle. We transform onto the reference triangle, Z Z 1 2 1 u ds = (u ◦ FT )2 ds, R r r VE R E1 E VE 1

and write uR := u ◦ FT . Due to symmetry, we consider only the right half of the 1 edge E1R , where rE1R = 1−x , and finally apply a trace inequality: 1 Z 1 1 uR (x1 , 0)2 dx1 1 − x 1 0 Z 1 Z 1−x1 h ∂uR i2 1 1 [uR ]2 dx2 dx1 (1 − x1 ) + ∂x2 1 − x1 0 1 − x1 0 |||uR |||2T R ' |||u|||2T . This leads us immediately to Theorem 8. Let u2 be as in Theorem 5. Then, the decomposition X (9) u2 = ΠE 0 u2 + u3 E∈Ef

satisfies u3 = 0 on (10)

S

E∈Ef

X E∈Ef

E and is bounded in the sense of 2 2 2 k∇ΠE 0 u2 kL2 + k∇u3 kL2 |||u2 ||| .

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3.4. Main result. Proof of Theorem 1 for the case of triangles. Summarizing the last subsections, we have X X ΠV0 u1 , u3 = u2 − u1 = u − Πh u, u 2 = u1 − ΠE 0 u2 , V ∈Vf

E∈Ef

and the decomposition (11)

u = Πh u +

X

ΠV0 u1 +

V ∈Vf

X

ΠE 0 u2 +

E∈Ef

X

u3 |T .

T ∈T

is stable in the k · kA -norm. For any edge E or triangle T , we can find a vertex V , such that the corresponding summand is in VV . Since for each vertex only finitely many terms appear, we can use the triangle inequality and finally arrive at the missing spectral bound X 2 hCu, ui = inf ku k kuV k2A hAu, ui . + 0 A P u=u0 + V uV u0 ∈V0 ,uV ∈VV

V

4. Sub-space splitting for tetrahedra Most of the proof for the 3D case follows the strategy introduced in Section 3, so we use the definitions thereof. The only principal difference is the edge interpolation operator, which shall be treated in more detail. 4.1. Coarse and vertex contributions. We define the level surfaces of the vertex hat basis functions ΓV (x) := ΓV (ϕV (x)) := {y : ϕV (y) = ϕV (x)}. As in 2D, we first subtract the coarse grid function u1 = u − Πh u, and secondly the multi-dimensional vertex interpolant to obtain u 2 = u 1 − ΠV u 1 , where the definitions of ΠV , ΠV0 , ΠV are the same as in Section 3, only the level set lines γV are replaced by the level surfaces ΓV . With the same arguments, one easily shows that X (12) kΠV0 u1 k2A + k∇u2 k2L2 + krV−1 u2 k2L2 kuk2A . v∈Vf

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13 VE,1

t γE(0,0)

1

E

R

γE(x)

x E

F 0

1

s

Figure 3. Reference triangle

VE,2

Figure 4. Edge patch

4.2. Edge contributions. Let F := {(s, t) : s ≥ 0, t ≥ 0, s + t ≤ 1} be the reference triangle in Figure 3. For (s, t) ∈ F , we define the level lines γE (s, t) := {x : ϕVE,1 (x) = s and ϕVE,2 (x) = t}, and write γE (x) := γE (ϕVE,1 (x), ϕVE,2 (x)) for the level line corresponding to a point x in the edge-patch ωE , see Figure 4. Define the space of constant functions on these level lines, SE := {v : v|γE (x) = const} and its polynomial subspace SE,p := SE ∩ Vp . The edge averaging operator into SE reads Z 1 E Π v (x) := v(y) dy. |γE (x)| γE (x) Furthermore, let rVE := min{rVE,1 , rVE,2 }, and rE (x) := dist{x, E}. Lemma 9. The edge-averaging operator satisfies (i) ΠE Vp = SE,p , (ii) (ΠE u)(x) = u(x), for x ∈ E, u continuous, (iii) k∇ΠE ukL2 (ωE ) k∇ukL2 (ωE ) , (iv) krV−1 ΠE ukL2 (ωE ) krV−1 ukL2 (ωE ) , E (v) krE−1 (u − ΠE u)kL2 (ωE ) k∇ukL2 (ωE ) , where u ∈ H 1 (ωE ).


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The proof is analogous to the proofs of Lemma 3 and Lemma 4. Next, the edge-interpolation operator is modified to satisfy zero boundary conditions on ∂ωE . By the isomorphism (13)

for v ∈ SE ,

vF (s, t) := v|γE (s,t) ,

the function space SE can be identified with a space on the triangle F . Lemma 10. The isomorphism (13) fulfills the following equivalences for functions v ∈ SE : (i) 1/2 kvkL2 (ωE ) ' h3/2 krE R vF kL2 (F ) , (ii) 1/2 k∇vkL2 (ωE ) ' h1/2 krE R ∇vF kL2 (F ) , (iii)

1/2

r −1 1/2 E R , vF krVE vkL2 (ωE ) ' h

rV ER

L2 (F )

(iv) −1/2

krE−1 vkL2 (ωE ) ' h1/2 krE R vF kL2 (F ) , where E R := {(s, t) ∈ F : s + t = 1}, rE R (s, t) := 1 − s − t,

and

(rVER )−1 :=

1 1−s

+

1 . 1−t

Proof. We parameterize the edge-patch ωE by FE : γE (0, 0) × F → ωE (z, (s, t)) 7→ z + s(VE,1 − z) + t(VE,2 − z). Note that functions v ∈ SE do not depend on the parameter z ∈ γE (0, 0) and vF (s, t) = (v ◦ FE )(z, s, t) for any z ∈ γE (0, 0). Equivalence (i) holds due to the transformation of the integrals Z Z Z 1 Z 1−s 2 2 |v| dx ' h |v ◦ FE |2 (1 − s − t) dt ds dz ωE γ (0,0) 0 0 ZE |vF |2 rE R d(s, t). ' h3 F

Derivatives evaluate to

∂vF ∂s

= (∇v) ·

∂FE ∂s

= (∇v) · (VE,1 − z), and thus

|(∇v) ◦ FE | ' h−1 |∇vF |. In combination with (i), we have proven (ii). Finally, equivalences (iii) and (iv) follow from rVE ◦ FE ' h rVER .

P-VERSION ASM

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(s,t)

Figure 5. Averaging lines of the smoothing operator Ss We now modify the function uF (s, t) := (ΠE u2 )|γE (s,t)

(14)

to obtain a function uF,00 which satisfies zero boundary conditions on the edges s = 0 and t = 0, and coincides with uF on the edge s + t = 1. This modification is done in such a way that it is continuous in the weighted H 1 -norm. First, we define the smoothing operator (cf. Figure 5) Z 1 τ (Ss v)(s, t) := v s + (1 − s − t), t dτ. 2 0 Secondly, we modify the operator to obtain (Ss,0 v)(s, t) := (Ss v)(s, t) −

1−s−t (Ss v)(0, t), 1−t

which vanishes on the edge s = 0. Lemma 11. The smoothing operator Ss,0 satisfies Ss,0 : {v ∈ P p : v(0, 1) = v(1, 0) = 0} → {v ∈ P p : v(1, 0) = v(0, ·) = 0} and the estimates

1/2

r1/2R

−1/2

r R ∇(Ss,0 v)

E (Ss,0 v) + + rE R (Ss,0 v − v) L2 (F ) rV E L2 (F ) L2 (F ) ER

(15)

1/2

r1/2 rE R ∇v L2 (F ) + rVER v L2 (F ) . ER

Proof. Assume that v is a polynomial vanishing in (0, 1) and (1, 0). Then Ss v is again a polynomial, which is identical to v on the edge s + t = 1. In particular, the restriction onto the edge s = 0 is a polynomial in t vanishing for t = 1. Thus, the factor 1 − t in the definition of Ss,0 cancels out.


16

First, we prove the corresponding estimates for the smoothing operator Ss . We observe that derivatives of Ss v depend on derivatives of v, only: Z 1 ∂(Ss v) τ = (∇v) · 1 − , 0 dτ, ∂s 2 0 Z 1 ∂(Ss v) τ = (∇v) · − , 1 dτ. ∂t 2 0 Since rE R is bounded from below and from above on the averaging line [(s, t); (s+ 1 (1 − s − t), t)], the smoothing operator Ss is bounded in the weighted H 1 2 semi-norm. The approximation property corresponding to the weighted L2 -norm follows from Friedrichs’ inequality applied on the same line. Now, we prove the estimates for the correction Ss,0 − Ss . The first is

i h

1/2 1−s−t (Ss v)(0, t)

rE R ∇ 1−t L2 (F )

1/2 −1 −s

≤ rE R , (S v)(0, t)

s 2 1−t (1−t)

L2 (F )

1/2 1−s−t

+ rE R ∇(Ss v)(0, t) 1−t

L2 (F )

∂(Ss v) (0, t) L2 (F ) (1 − t)−1/2 (Ss v)(0, t) L2 (F ) + (1 − t)1/2 ∂t

∂(S v) s = (Ss v)(0, t) L2 (0,1) + (1 − t) (0, t) L2 (0,1) , ∂t the other two are bounded by the same expression. We now bound these trace norms of Ss v by the right-hand side of (15). We start with the L2 -norm: 1

Z

Z

2

(Ss v) (0, t) dt = 0

1

hZ

v

1 − t

i2 τ, t dτ dt

2 Z 1 Z 1−t 1 2 2 2 1−t v τ, t dτ dt = ds dt v 2 (s, t) 2 1−t 0 0 0 0

Z

1

0

1Z

≤ Z0 F

rE R 2 v (s, t) ds dt rV2 E

We have substituted s =

s≤

1−t 2

1−t 2

τ , and used that

implies

1 1−s−t r R ≤ 2 E2 . ≤2 2 1−t (1 − t) rVE

P-VERSION ASM

17

Similarly, we can bound the weighted H 1 -norm on the edge by Z 1 Z 1 h i2 hZ 1 i2 2 ∂(Ss v) 2 (1 − t) (0, t) dt = (1 − t) (∇v) · (−τ /2, 1)T dτ dt ∂t 0 0 0 Z 1 Z 1 2 1 − t τ, t dτ dt (1 − t)2 ∇v 2 0 0 Z 1 Z 1−t 2 (1 − s − t)|∇v|2 ds dt. 0

0

In the same manner, we define Z (St u)(s, t) := 0

1

τ u s, t + (1 − s − t) dτ, 2

and

1−s−t (St uF,0 )(s, 0). 1−s These two smoothing operators allow us to define the function (St,0 )(x, y) := (St uF,0 )(s, t) −

uF,00 := St,0 Ss,0 uF satisfying zero boundary values at both edges s = 0 and t = 0. We define the edge interpolation operator by (16)

(ΠE 0 u2 )(x) := uF,00 (ϕVE,1 (x), ϕVE,2 (x)).

Lemma 12. The edge interpolation operator ΠE 0 satisfies −1 krE−1 {v − ΠE 0 v}kL2 (ωE ) k∇vkL2 (ωE ) + krV vkL2 (ωE ) ,

where rE (x) := min rE (x). E∈E

Proof. The proof is analogous to the one of Lemma 4, part (v). We observe that X krE−1 wkL2 (ωE ) ' krE−1 wkL2 (ωE ) + krE−10 wkL2 (ωE ) . E 0 ⊂ωE \{E}

The desired estimate for the first term follows directly from Lemma 9, Lemma 10 and Lemma 11. R For the second term, we use that T rE−20 v 2 dx k∇vk2L2 (T ) for functions v vanishing on ∂ωE . Finally, we define the global edge interpolation operator X (17) ΠE := ΠE 0, E∈Ef

where Ef is the set of are all free edges, i. e. those which do not lie completely on the Dirichlet boundary. We obtain

18


Theorem 13. The decomposition u2 =

(18)

X

ΠE 0 u2 + u3

E∈Ef

fulfills the stability estimate X −1 −1 2 2 2 2 2 kΠE (19) 0 u2 kA + k∇u3 k + krE u3 k k∇u2 k + krV u2 k . E∈Ef

Moreover, u3 = 0 on

S

E∈Ef

E.

Proof. The result is an immediate consequence of Lemma 9, Lemma 11 and Lemma 12 using the argument of finite summation. 4.3. Main result. Proof of Theorem 1 for the case of tetrahedra. The interpolation on faces in 3D and its analysis follows the line of the edge interpolation in 2D, see Section 3.3. Summarizing, we obtain X ΠV0 u1 , u1 = u − Πh u, u2 = u 1 − V ∈Vf

u3 = u2 −

X

ΠE 0 u2 ,

u 4 = u3 −

X

ΠF0 u3 ,

F ∈Ff

E∈Ef

where Ff = {F ∈ F : F 6⊂ ΓD }. As a consequence of the last subsections, the decomposition X X X X F (20) u = Πh u + ΠV0 u1 + ΠE Π u4 |T u + u + 2 3 0 0 V ∈Vf

F ∈Ff

E∈Ef

is stable in the k · kA -norm.

T ∈T

5. Numerical results

In this section, we show numerical experiments on model problems to verify the theory elaborated in the last sections and to get the absolute condition numbers hidden in the generic constants. Furthermore, we study two more preconditioners. We consider the H 1 (Ω) inner product A(u, v) = (∇u, ∇v)L2 + (u, v)L2 on the unit cube Ω = (0, 1)3 , which is subdivided into 69 tetrahedra, see Figure 6. We vary the polynomial order p from 2 up to 10. The condition numbers of the preconditioned systems are computed by the Lanczos method. Example 1: The preconditioner is defined by the space-decomposition with big overlap of Theorem 1: X V = V0 + VV V ∈V

P-VERSION ASM

19

The condition number is proven to be independent of h and p. The computed numbers are drawn in Figure 7, labeled ’overlapping V’. The inner unknowns have been eliminated by static condensation. The memory requirement of this preconditioner is considerable: For p = 10, the memory needed to store the local Cholesky-factors is about 4.4 times larger than the memory required for the global matrix. In Section 2 we introduced the space splitting into the coarse space V0 and the vertex subspaces VV . However, our proof of Theorem 1 involves the finer splitting of a function u into a coarse function, functions in the spider spaces SV , edge-, face-based and inner functions. Other additive Schwarz preconditioners with uniform condition numbers are induced by this finer splitting. Example 2: Now, we decompose the space into the coarse space, the pdimensional spider-vertex spaces SV,0 = span{ϕV , . . . , ϕpV }, and the overlapping sub-spaces VE on the edge patches: X X VE SV,0 + V = V0 + V ∈V

E∈E

The condition number is proven to be uniform in h and p. The computed values are drawn in Figure 7, labeled ’overlapping E, spider V’. Storing the local factors is now about 80 percent of the memory for the global matrix. Example 3: The interpolation into the spider-vertex space SV,0 has two continuity properties: It is bounded in the energy norm, and the interpolation rest satisfies an error estimate in a weighted L2 -norm, see Lemma 4 and equation (12). Now, we reduce the p-dimensional vertex spaces to the spaces spanned by the low energy vertex functions ϕl.e. V defined as solutions of min v∈SV,0 , v(V )=1

kvk2A .

These low energy functions can be approximately expressed by the standard vertex functions via ϕl.e. = f (ϕV ), where the polynomial f solves a weighted V 1D problem and can be given explicitly in terms of Jacobi polynomials, see the upcoming report [BPP05]. The interpolation to the low energy vertex space is uniformly bounded, too. But, the approximation estimate in the weighted L2 norm depends on p. The preconditioner is now generated by X X V = V0 + span{ϕl.e. } + VE . V V ∈V

E∈E

The computed values are drawn in Figure 7, labeled ’overlapping E, low energy V’, and show a moderate growth in p. Low energy vertex basis functions obtained by orthogonalization on the reference element have also been analyzed in [Bic97, SC01].


20

Figure 6. Unstructured mesh

3

160

overlapping V overlapping E, spider V overlapping E, low energy V

standard vertex 140

2.5

condition number

condition number

120 2

1.5

1

100 80 60 40

0.5 20 0 2

4

6 polynomial order

8

0

10

2

Figure 7. Overlapping blocks

4

6 polynomial order

8

10

Figure 8. Standard vertex

Example 4: We also tested the preconditioner without additional vertex spaces, i.e., X V = V0 + VE . E∈E

Since vertex values must be interpolated by the lowest order functions, the condition number is no longer bounded uniformly in p. The rapidly growing condition numbers are drawn in Figure 8. References [Ain96a] M. Ainsworth. Hierarchical domain decomposition preconditioner for h-p finite element approximation on locally refined meshes. SIAM J. Sci. Comput, 17:1395–1413 (1996)

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21

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[SC01] S. J. Sherwin and M. Casarin. Low Energy Basis Preconditioning for Elliptic Substructured Solvers Based on Unstructured Spectral/ hp Element Discretisations. Journal of Computational Physics, 171:1–24 (2001) [SBG96] B. F. Smith, P. E. Bjørstad, and W. D. Gropp. Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press (1996) [SB91] B. Szab´ o and I. Babuˇska. Finite Element Analysis. Wiley, (1991) [SDR04] B. Szab´ o, A. D¨ uster, E. Rank. The p-version of the finite element method. In E. Stein, R de Borst, T.J.R. Hughes, eds, Encyclopedia of Computational Mechanics, John Wiley & Sons, (2004) [TW04] A. Toselli and O. Widlund. Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics, Vol. 34 (2005) [Qua99] A. Quarteroni and A. Valli Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999) ¨ berl, Johann Radon Institute for Computational and Applied MatheJ. Scho matics (RICAM), Austria, [email protected] J. M. Melenk, The University of Reading, Department of Mathematics, UK, [email protected] C. G. A. Pechstein, Institute for Computational Mathematics, Johannes Kepler University Linz, Austria, [email protected] S. C. Zaglmayr, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria, [email protected]