The results are summarized in Tables 1 and 2, where niter and cond denote the ... MGDD method is used to solve the problem (4.9), it takes niter = 22 outer ...
SUBSTRUCTURING PRECONDITIONING FOR FINITE ELEMENT APPROXIMATIONS OF SECOND ORDER ELLIPTIC PROBLEMS. II. MIXED METHOD FOR AN ELLIPTIC OPERATOR WITH SCALAR TENSOR SERGUEI MALIASSOV
Abstract. This work continues the series of papers in which new approach of constructing algebraic multilevel preconditioners for mixed nite element methods for second order elliptic problems with tensor coecients on general grid is proposed. The linear system arising from the mixed methods is rst algebraically condensed to a symmetric, positive de nite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming nite element methods. Algebraic multilevel preconditioners are then constructed for this system based on a triangulation of parallelepipeds into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed nite element methods are presented to illustrate the present theory. Key words. Mixed method, nonconforming method, multilevel preconditioner, condition number, second order elliptic problem AMS(MOS) subject classi cations. 65N30, 65N22, 65F10
1. Introduction. Let be a bounded domain in IR3 , with the polygonal boundary
@ . We consider the elliptic problem ?r (aru) = f in ; (1.1) u=0 on @ ; where a(x) is a uniformly positive de nite, bounded, symmetric tensor and f (x) 2 L2 ( ). Let ( ; )S denote the L2 (S ) inner product (we omit S if S = ), and let V = H (div; ) = fv 2 (L2 ( ))3 : r v 2 L2 ( )g; W = L2 ( ): Then (1.1) is formulated in the following mixed form for the pair (q; u) 2 V W : (r q; w) = (f; w); 8w 2 W; (1.2) ? 1 (a q; v) ? (u; r v) = 0; 8v 2 V: It can be easily seen that (1.1) is equivalent to (1.2) through the relation q = ?aru: In applications of uid ow in porous media, u(x) is referred to as pressure and q as to Darcy velocity vector. It is well known that (1.2) has a unique solution u(x) 2 H01( ) \ H 2( ), and that the following elliptic regularity estimate holds true (cf. [14]): kuk2; ckf k0; ; Institute for Scienti c Computation and Department of Mathematics, Texas A&M University, 326 Teague Research Center, College Station, TX 77843{3404. 1
2
S. Maliassov
where c is a constant dependent only on and where k k0; and k k2; are the L2 ( ) and H 2( ) Sobolev norms, respectively de ned by
Z
2 kuk0; = u dx ; 1 2
0Z 1 X kuk2; = @
j@ uj2 dxA : 1 2
j jm
To de ne a nite element method, we need a partition Th of into elements T , say, simplexes, rectangular parallelepipeds, and/or tetrahedra. In Th, we also need that adjacent elements completely share their common edge or face; let @ Th denote the set of all interior faces e of Th. Let V~ h Wh V W denote some standard mixed nite element space for second order elliptic problems de ned over Th (see, e.g., [5], [6], [11], [21], and [22]). This space is nite dimensional and de ned locally on each element T 2 Th, so let Vh (T ) = Vh jT and Wh(T ) = WhjT . Then the mixed nite element method for (1.1) is to nd (qh; uh) 2 V~ h Wh: (r qh; w) = (f; w); (a?1qh; v) ? (uh ; r v) = 0;
(1.3)
8w 2 Wh; 8v 2 V~ h:
The requirement V~ h V implies that the normal component of the vector q is continuous across the interelement boundaries @ TT . Following [2], we relax this constraint on V~ h by de ning Vh = fq 2 (L2h( ))3 : qjT 2 V~ h(T ) for each T 2 Thg. In order to enforce the interelement continuity of the normal component of q we need to introduce the space of the Lagrange multipliers
9 8 [ = < e : je 2 V~ h e for each e 2 @ Th ; ; Lh = : 2 L2 e2@ Th
where is the unit normal to e. Also, to establish a relationship between the mixed method and the nonconforming Galerkin method and to construct ecient preconditioners, following [9] and [10] we introduce the projection of the coecient, i.e., h = Pha?1 , where Ph is the L2 -projection onto Wh. Then the hybrid form of the mixed method for (1.1) is to nd (qh; uh; h) 2 Vh Wh Lh such that
P (r q ; w) = (f; w); h T
T 2Th
(1.4)
P (uh; r v)T ? (h; v T )@T n@ = 0; ( hqh; v) ? T 2Th
P (q ; ) h T @T n@ = 0;
T 2Th
8w 2 Wh; 8v 2 Vh; 8 2 Lh:
Note that the last equation in (1.4) enforces the continuity requirement mentioned above, so in fact qh 2 V~ h. In [2] and [20], it was shown that the solution to (1.4) can be obtained from a certain modi ed nonconforming Galerkin method by means
Substructuring Preconditioners For Mixed Methods
3
of augmenting the latter with bubble functions. Such a relationship has been studied recently for a large variety of mixed nite element spaces [1, 8, 9]. In this paper, following [10] it is shown that the linear system generated by (1.4) can be algebraically condensed to a symmetric, positive de nite system for the Lagrange multiplier h. It is then shown that this linear system can be obtained from the standard nonconforming Galerkin method without using any bubbles. The main objective of this paper is to construct algebraic multilevel preconditioners for the mixed nite element method. We rst use the above equivalence to construct multilevel preconditioners for the linear system for the Lagrange multipliers. Then the mixed method solutions qh and uh are recovered via these multipliers. The construction of multilevel preconditioners for the mixed methods is inspired by the fundamental work [4], [16], where new systematic representations for preconditioners in the Neumann-Dirichlet domain decomposition methods for conforming nite elements were suggested. The multilevel domain decomposition versions of these methods were outlined in detail in [17, 18]. In addition, the superelement approach used here to estimate condition numbers for two level methods is based on that used in [3, 12, 17, 19]. A detailed description of procedures to construct such preconditioners can be found in [10, 12, 13]. In all these works authors de ned partitioning Th of the whole domain subdividing it into topological parallelepipeds and splitting each parallelepiped in turn into six tetrahedra. The present paper prolongates these results to the case of splitting each topological parallelepiped into ve tetrahedra. Brie y, the approach used here to construct preconditioners includes two main stages. First, using the idea of partitioning (decomposing) a parallelepiped grid into tetrahedral substructures a three-level preconditioner is constructed with a \7-point" algebraic system on the coarse level, and the condition number of the preconditioned matrix is estimated. The explicit bounds of spectrum of the preconditioned matrix are obtained with help of the superelement approach [12, 17]. On the second stage, we de ne the preconditioner for the above 7-point algebraic system with one unknown per parallelepiped and show that this preconditioner is equivalent to the standard nite element approximation of the equation (1.1) with modi ed coecient tensor a~(x). To solve this problem we can use any well known technique. Namely, in this paper we use multilevel domain decomposition method [16, 17] to solve this auxiliary coarse level problem. The constructed preconditioners are spectrally equivalent to the original sti ness matrix and their arithmetic cost does not depend on the mesh size h and jump of the coecient a(x). Explicit estimates of condition numbers are obtained for these multilevel preconditioners. A computational scheme for implementing these preconditioners is also considered, and a three-step preconditioned conjugate gradient method using the present technique is described as well. In this paper the case where a(x) is a scalar tensor and is a regular parallelepiped is analyzed in detail for the three-level and multilevel preconditioners. The rest of the paper is organized as follows. In the next section we consider an elimination procedure for (1.4). Then, in section 3 we develop three-level preconditioner
4
S. Maliassov
for the resulting linear system. Development of three-level preconditioner to multilevel one and the multilevel domain decomposition method for solving coarse level problem is described in section 4. Finally, in section 5 extensive numerical results are presented for both nonconforming and mixed methods on logical parallelepipeds to illustrate the present theory.
2. The mixed nite element method. We now consider the most useful partition Th of into tetrahedra. In this section we outline the elimination procedure for
the system (1.4). Detailed description can be found in [1, 10]. The lowest-order Raviart-Thomas-Nedelec space [22, 21] de ned over T 2 Th is given by
Vh(T ) = (P0(T ))3 ((x; y; z)P0(T ));
Wh(T ) = P0(T ); Lh (e) = P0(e);
where Pi(T ) is the restriction of the set of all polynomials of total degree not bigger than i 0 to the set T 2 Th. For each T in Th, let fT = jT1 j (f; 1)T , where jT j denotes the volume of T . Also, set h = ( ij ) and qhjT = (qT 1 ; qT 2; qT 3) = (rT1 + tT x; rT2 + tT y; rT3 + tT z). Then, by the rst equation of (1.4) it follows that
tT = fT =3:
(2.1)
Now, take v = (1; 0; 0) in T and v = 0 elsewhere, v = (0; 1; 0) in T and v = 0 elsewhere, and v = (0; 0; 1) in T and v = 0 elsewhere, respectively, in the second equation of (1.4) to obtain (2.2)
(
3 X
i=1
jiqTi; 1)T +
4 X
i=1
jeiT j Ti(j) hje = 0; j = 1; 2; 3; i T
where jeiT j is the area of the face eiT , and Ti = (Ti(1) ; Ti(2); Ti(3)). Let T = ( ijT ) = (( ij ; 1)T )?1. Then (2.2) can be solved for rTj : (2.3)
4 rTj = ? P jeiT j jT1Ti(1) + jT2Ti(2) + jT3Ti(3) hjeiT ? i=1
3 ? f3 iP jiT ( i1x + i2y + i3z); 1 ; j = 1; 2; 3: =1 T
T
Let the basis in Lh be chosen as usual. Namely, take = 1 on one face and = 0 elsewhere in the last equation of (1.4). Then, apply (2.1) and (2.3) to see that the contributions of the tetrahedron T to the sti ness matrix and the right-hand side are
ATij
= iT T jT ;
FiT
(JTf ; iT )T + (JTf ; Ti )eiT ; =? jT j
T 2 Th ;
5
Substructuring Preconditioners For Mixed Methods
where iT = jeiT jTi and JTf = fT (x; y; z)=3. Hence we obtain the system for h:
A = F:
(2.4)
After the computation of h, we can recover qh via (2.1) and (2.3). Also, if uh is required, it follows from the second equation of (1.4) that 4 ! X 1 i uT = 3jT j ( qh; (x; y; z))T + hjeiT (x; y; z); T ei ; T i=1
T 2 Th :
The above result is summarized in the following lemma (see [10]). Lemma 2.1. Let Mh( ; ) = P ( ; T )@T T (; T )@T ; ; 2 Lh; T 2Th
Fh() = ? P
(J T 2Th jT j 1
f ; 1)T
(J f ; T )@T ; (; T )@T + TP 2T h
2 Lh ;
where J f is such that J f jT = JTf . Then h 2 Lh satis es
Mh(h; ) = Fh();
(2.5)
8 2 Lh;
where
Lh = f 2 Lh : je = 0 for each e @ g: Note that there are at most seven nonzero entries per row in the sti ness matrix A. Also, it is easy to see that the matrix A is a symmetric and positive de nite matrix; moreover, if the angles of every T in Th are not bigger than =2, then it is an M -matrix. Finally, (2.4) corresponds to the P1 nonconforming nite element method system, as described below. This equivalence is used to construct our multilevel preconditioners later. Following [10], let (2.6)
Nh = fv 2 L2 ( ) : vjT 2 P1(T ); 8T 2 Th; v is continuous
at the barycenters of interior faces and vanishes at the barycenters of faces on @ g:
Then the following proposition can be proved ([10]). Proposition 2.2. Let fh = Phf . Then (2.4) corresponds to the linear system produced by the problem: nd h 2 Nh such that (2.7) where ah ( h; ') =
P
ah( h; ') = (fh; '); ?1 T 2Th ( h r h; r')T .
8' 2 Nh;
6
S. Maliassov
3. Three level preconditioner over a cube. In this section we consider multi-
level preconditioners for (2.4) based on partitioning regular parallelepipeds into tetrahedral substructures, following the ideas in [12] and [13]. Here we treat the case where
is a unit cube and a(x) is a scalar tensor. Our goal is to introduce an algebraic formulation of the approximate problem using a type of static condensation that eliminates some of the unknowns. In this way we can reduce substantially the size of the problem. For this approach we need a special partitioning of the domain into tetrahedra that have some regularity and preserve the simplicity of the algebraic problem. Let Ch = fC (i;j;k)g be a partition of into uniform cubes with the length h = 1=n, where (xi ; yj ; zk ) is the right back upper corner of the cube C (i;j;k) . Next, each cube C (i;j;k) is divided into 5 tetrahedra as shown in Figure 1 and denote this partitioning of
into tetrahedra by TT . Let Wc;h be the space of piecewise constants associated with Ch, and Pc;h be the L2 -projection onto Wc;h. To de ne our preconditioner, we introduce h = Pc;ha?1 in the hybrid form (1.4) instead of h = Pha?1 . Obviously, Lemma 2.1 and Proposition 2.2 are still valid for this modi cation since Th is a re nement of Ch. With this modi cation, h?1 is a constant on each cube. For notational convenience, we drop the subscript h and simply write h?1 = ac diagf1; 1; 1g. Let Nh be the nonconforming nite element space associated with Th as de ned in (2.6), and let its dimension be N . All the unknowns on the faces of @ are excluded. For this reason N = 10n3 ? 6n2. For any function vh 2 Nh, we denote by v 2 IRN the corresponding vector of its degrees of freedom. Introduce the inner product (3.1)
(u; v)N = h3
X
pi 2@ Th
uh(pi)vh(pi);
8uh; vh 2 Nh;
where the pi's are the barycenters of the interior faces. It is easy to see that the norm induced by (3.1) is equivalent to the L2 -norm on . For each cube C = C (i;j;k) 2 Ch, denote by NhC the subspace of the restriction of the functions in Nh onto C . For each v 2 NhC , we indicate by vc its corresponding vector. The dimension of NhC is denoted by N C . Obviously, for a cube without faces on @ its dimension is 16, i.e., N C = 16. The local sti ness matrix AC on cube C 2 Ch is given by (3.2)
(AC uc ; vc)N C =
X
T C
( hruh ; rvh)T ;
8uh; vh 2 NhC :
Then the global sti ness matrix is determined by assembling the local sti ness matrices: (3.3)
(Au; v)N =
X
C 2Ch
(AC uc ; vc)N C ;
8u; v 2 IRN :
Now we consider a cube C that has no face on the boundary @ and enumerate the faces sj , j = 1; : : :; 16 of the tetrahedra in this cube as shown in Figure 2. Then the local sti ness matrix of this prism has the following form:
7
Substructuring Preconditioners For Mixed Methods
2 = ?= ?= ?= ? ? ? 3 66 ? = = ? = ? = 77 ? ? ? 66 ? = ? = = ? = 77 ? ? ? 66 ? = ? = ? = = ? ? ? 777 66 77 66 ? 77 66 ? 77 66 ? 77 6 ? 77 AC = 32h ac 66 ? 66 77 ? 66 77 66 77 ? 66 77 ? 66 77 ? 66 77 ? 64 75 ? ? 9 2
1 2
1 2
9 2
0
0
1 2
1 2 1 2
9 2
1 2
1
0
1
0
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
0
1
1
1
1
1
,
1
1
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
0
1
1
1
1
1
1
1
"
#
A11;c A12;c ; = 3h ac A 2 21;c A22;c
3 2 3 ? 1 ? 1 ? 1 66 ?1 3 ?1 ?1 77 1 A11;c = 3 I11;c + 2 66 ?1 ?1 3 ?1 77 ; 5 4
?1 ?1 ?1 3
0
1
0
1
0
AC
(3.5)
0
9 2
which we write as where
1
0
1 2
(3.4)
1
1 2
1 2
1 2
0
1
1 2
1 2
A22;c = I22;c :
Figure 1. Partition of cube into tetrahedra.
Along with matrix AC we also introduce the matrix B C as " # 3 h B 11;c A12;c C (3.6) ; B = 2 ac A A 21;c 22;c
8
S. Maliassov
where
2 3 3 ? 1 ? 1 ? 1 66 ?1 1 0 0 77 B11;c = 3 I11;c + 2 66 ?1 0 1 0 77 ; 4 5
?1 0
0
1
Proposition 3.1. It holds that kerAC = kerB C .
Proof. It is easy to see from the de nitions of AC and B C that kerAC = kerB C = fv = (v1; v2; ; v16)T 2 IR16 : vi = v1; i = 2; ; 16g. Remark. If the cube C 2 Ch has a face on @ , then the matrix AC does not have the rows and columns which correspond to the nodes on that face; the blocks A11;c and B11;c are the same as in the previous case and the modi cation of A22;c is obvious. We now de ne the N N matrix B by the following equality:
(B u; v)N =
(3.7)
X
C 2Ch
(B C uc ; vc)N C ;
8u; v 2 IRN :
Since the matrix B is used for preconditioning the original problem (2.4), it is important to estimate the condition number of B ?1A. Lemma 3.2. Let c satisfy the equality
AC u c = c B C u c ;
(3.8)
C 2 Ch :
Then we have
(3.9)
(Au; u)N max and max C 2Ch c (Bu;u)N 6=0 (B u; u)N
(Au; u)N min : min C 2Ch c (Bu;u)N 6=0 (B u; u)N
Proof. For each C 2 Ch, it follows from (3.8) that
(AC uc ; uc)N C = c (B C uc ; uc)N C : It then follows from the fact that all local sti ness matrices are nonnegative that
X
C 2Ch
X
c (B C uc ; uc)N C C 2Ch X C (B uc ; uc)N C : max c C 2C
(AC uc ; uc)N C =
h
C 2Ch
Hence from the de nitions of A and B , we see that (Au; u)N max (B u; u)N : C 2C c h
Consequently, the rst inequality in (3.9) is true. The same argument can be used to show the second inequality. From Lemma 3.2, we see that, to estimate the condition number of B ?1A, it suces to consider the local problems (3.8). Using a superelement analysis [16, 19], to estimate
9
Substructuring Preconditioners For Mixed Methods
13
12
11
7
8
@ @ @ x @ 3 @@ 1 @ 2 @ @ @ @ 4@ @ @ 5 @ @@ @ @ @ @ 14 @ 10 @ 16 @ z
y
15
6
9
(a) Cube of type I
10
9
1
z
x
y
?
4
6 8
?
16
?
?
?
? 11
?
2
? ?
? 7
?
?
?
?
?
5
? 3
13
14
15
12
(b) Cube of type II Figure 2. Local enumeration of faces in cubes.
10
S. Maliassov
max and Cmin , it suces to treat the worst case where the cube C 2 Ch has no C 2Ch c 2Ch c face on the boundary @ . From (3.4) and (3.6), a direct calculation shows that the eigenvalues c are within the interval [1=4; 1]. Then the inequalities (3.9) yield: Proposition 3.3. The eigenvalues of problem
Au = B u
(3.10)
belong to the interval [1=4; 1] and the condition number is thus estimated by
cond(B ?1A) 4: We stress that the condition number of the matrix B ?1A is bounded by a constant independent of the step size of the mesh h and the jump of the coecient a(x). Since we introduced a two level subdivision, the matrix B can be referred to as a two level preconditioner. Then, in this section we propose a modi cation of the matrix B and consider its properties. Toward that end, we divide all unknowns in the system into three groups: 1. The rst group consists of the one unknown per cube corresponding to the 1st faces of the tetrahedra that are internal for each cube C 2 Ch (see Figure 2, faces 1). 2. The second group consists of all unknowns corresponding to faces of the cubes in the partition Ch, excluding the faces on @ (Figure 2, faces 5; 6; : : :; 16). 3. The third group consists of the unknowns corresponding to the faces of the tetrahedra that are internal for each cube and which are not in the 1st group (these are unknowns on faces 2, 3 and 4 on Figure 2). This splitting of the space IRN induces the presentation of the vectors: vT = (v1T ; v2T ), where v1 2 IRN and v2 2 IRN , where v2 corresponds to the unknowns of the 3-rd group. Obviously, N2 = 3n3 and N1 = N ? 3n3. Then the matrix B can be presented in the following block form: 1
(3.11)
2
"
# B 11 B12 B= B B ; 21 22
dimB11 = N1 :
Denote now by B^11 = B11 ? B12 B22?1B21 the Schur complement of B obtained by elimination of the vector v2. Then B11 = B^11 + B12 B22?1 B21, so the matrix B has the form " ^ ?1 B B # B + B B 11 12 12 : 22 21 B= (3.12) B21 B22 Note that for each cube C 2 Ch the unknowns of the 3rd group (unknowns on the faces 2, 3 and 4 in local enumeration, see Figure 2) are connected only with the unknowns of the 1st and 2nd groups and therefore the matrix B22 is diagonal and can be
Substructuring Preconditioners For Mixed Methods
11
inverted locally (cube by cube). Thus, matrix B^11 is easily computable. The proposed modi cation of the matrix B from (3.12) is of the form " ~ ?1 B B # B + B B 0 12 12 22 21 B~ = B21 B22 ; where B~0 is to be de ned later. Consider now the restriction of the matrix B on a single cube
BC
"
# B 11;c B12;c = B ; 21;c B22;c
and de ne the Schur complement on a cube by B^11;c = B11;c ? B12;c B22?1;c B21;c . In the local enumeration introduced on Figure 2 the matrix B^11;c has the form
2 33=5 ?1 ?1 ?1 ?2=5 ?2=5 ?2=5 ?2=5 ?2=5 ?2=5 ?2=5 ?2=5 ?2=5 3 66 ?1 1 77 66 ?1 77 1 66 ?1 77 1 66 ?2=5 77 4=5 ?1=5 ?1=5 66 ?2=5 77 ?1=5 4=5 ?1=5 3 h ^ 6 77 B11;c = 2 ac 6 ?2=5 ?1=5 ?1=5 4=5 66 ?2=5 77 4=5 ?1=5 ?1=5 66 ?2=5 77 ?1=5 4=5 ?1=5 66 ?2=5 77 ?1=5 ?1=5 4=5 66 ?2=5 4=5 ?1=5 ?1=5 77 4 ?2=5 ?1=5 4=5 ?1=5 5 ?2=5
?1=5 ?1=5 4=5
Remark. If the cube C 2 Ch has a face on @ , then the matrix B^11;c does not have the rows and columns which correspond to the nodes on that face. Following [12], we introduce on each cube a modi cation of the matrices B^11;c in the form: 2 12 ?1 ?1 ?1 ?1 ?1 ?1 ?1 ?1 ?1 ?1 ?1 ?1 3 66 ?1 1 77 66 ?1 77 1 66 ?1 77 1 66 ?1 77 1 66 ?1 77 1 77 B~11;c = 32h ac 66 ?1 1 66 ?1 77 1 66 ?1 77 1 66 ?1 77 1 66 ?1 77 1 4 ?1 5 1 ?1 1 Proposition 3.4. The matrices B^11;c and B~11;c have the same kernel, i.e., kerB^11;c =
kerB~11;c .
12
S. Maliassov
Proof. It can be easily checked that kerB^11;c = kerB~11;c = fv = (v1; v2; ; v13)T 2 IR13 : vi = v1; i = 2; ; 13g. We now consider the eigenvalue problem
B^11;c u = B~11;cu;
(3.13)
u 2 IR13 :
It is easy to check the proposition Proposition 3.5. The eigenvalues of problem (3.13) belong to the interval [2=5; 1]. Now de ning a new matrix on each cube: " ~ # ?1 B B + B B B 11 ;c 12 ;c 21 ;c 12 ;c C 22;c B~ = (3.14) B21;c B22;c : we de ne the symmetric positive-de nite N1 N1 matrix B~0 by (B~0u1 ; v1) =
X ~ (B11;c u1;c ; v1;c);
C 2Ch
where v1 ; u1 2 IRN , and u1;c and v1;c are their respective restrictions on the cube C . As in (3.12), we introduce the matrix 1
(3.15)
" ~ ?1 B B # B + B B 0 12 21 12 22 B~ = B21 B22 :
Using Propositions 3.3 and 3.5, and the same proof as in Proposition 3.2 we have the following theorem. Theorem 3.6. The matrix B~ de ned in (3.15) is spectrally equivalent to the matrix A, i.e., ~ B~ A B; where = 1=10 and = 1. Moreover,
(3.16)
cond(B~ ?1A) = 10:
Instead of the matrix B in the form (3.12) we take the matrix B~ from (3.15) as a preconditioner for the matrix A. As we noted earlier, the matrix B22 is block-diagonal and can be inverted locally on cubes. So we concentrate on the linear system (3.17)
B~0u = G:
In terms of the group partitioning in section 3, the matrix B~0 has the block form (3.18)
"
# C C 11 12 B~0 = C C ; 21 22
Substructuring Preconditioners For Mixed Methods
13
where the block C22 corresponds to the nodes from the second group, which are on the faces of tetrahedra perpendicular to the coordinate axes. From the de nition of B~0 , it can be seen that the matrix C22 is diagonal. In the above partitioning, we present u and G in (3.17) in the form
"
#
"
u = uu12 ;
(3.19)
#
1 G= G G2 :
Then, after elimination of the second group of unknowns:
u2 = C22?1(G2 ? C21 u1); we get the system of linear equations (3.20)
~ 1; (C11 ? C12 C22?1C21 )u1 = G1 ? C12C22?1 G2 G
where the vector u1 and the block C11 correspond to the unknowns from the rst group, which have only one unknown per each cube. The dimension of vectors u1 and G1 is obviously equal to
M = dim(u1) = n3: Thus, de ning as above Schur complement of matrix B~0 by C^11 = C11 ? C12C22?1C21 matrix B~ can be presented in the form (3.21)
# 2" ^ 3 C11 + C12 C22?1C21 C12 + B B ?1B B 12 22 21 12 7 B~ = 64 5; C21 C22 B21 B22
where matrices B22 and C22 are diagonal and can be inverted locally cube-by-cube. Again, we have to stress that the condition number of the matrix B~ ?1 A is bounded by the constant independent of the step size of the mesh h and the jump of the coecient a(x). The matrix B~ can be referred to as a three-level preconditioner. By making straightforward calculations it can be shown that the Schur complement ^ C11 is \7-point-scheme" matrix. Introducing for each cube C (i;j;k) the coecients
K (3.22)
(i;j;k) 1
=
K2(i;j;k) = K3(i;j;k) =
!
3h 2 2 ! 3h 2 2 ! 3h 2 2
a(i;j;k) a(i+1;j;k) ; a(i;j;k) + a(i+1;j;k) a(i;j;k) a(i;j+1;k) ; a(i;j;k) + a(i;j+1;k) a(i;j;k) a(i;j;k+1) ; a(i;j;k) + a(i;j;k+1)
matrix C^11 can be schematically represented in the following form for C (i;j;k) \ @ = ;
14
S. Maliassov
[ ? K3(i;j;k)] [?K
"
(i;j ?1;k) ] 2
[ ? K1(i?1;j;k)]
K1(i?1;j;k) + K2(i;j?1;k) + K3(i;j;k?1) + +K1(i;j;k) + K2(i;j;k) + K3(i;j;k)
[ ? K1(i;j;k)]
#
[ ? K2(i;j;k)]
[ ? K3(i;j;k?1)]
If C (i;j;k) \ @ 6= ; then the previous scheme is modi ed in a natural way, for example for i = 1, j; k 6= 1; n, for unknown in cube C (1;j;k) we have the scheme [ ? K3(1;j;k) ]
2 3h (1;j;k) (1;j?1;k) (1;j;k?1) 3 + K2 + K3 +5 4 2 2 a (1;j;k ) (1;j;k) (1;j;k) +K1 + K2 + K3
[ ? K2(1;j?1;k) ]
[ ? K1(1;j;k) ]
[ ? K2(1;j;k)]
[ ? K3(1;j;k?1) ]
In the next section we consider the solution techniques for problem (3.20) with the matrix C^11 :
C^11 v = g:
(3.23)
4. Multilevel preconditioner over a cube. While the preconditioner B~ has
good properties, it is still not economical to invert it because the entries of the matrix C^11 depend on jump of the coecients. In this section we propose a modi cation of the matrix C^11 provided additional assumptions on the behavior of the function a(x) and show that for that modi cation we can use any well known multilevel procedure. Assumption (A1): Suppose that unit cube can be represented as a union of a certain number m of pairwise disjoint cubes Gi, i = 1; : : :; m with the size of edge H (H > 2h) in such a way that in each cube Gi the function a(x) is a positive constant. m In other words, we set = S G i and a(x) = consti > 0, x 2 Gi, i = 1; : : :; m. i=1 ( i;j;k ) For each cube C 2 TC consider the following submatrices (4.1)
(i;j;k)
Sl
(i;j;k)
= Kl
"
#
1 ?1 ?1 1 ;
l = 1; 2; 3; i; j; k = 2; : : :; n ? 1;
15
Substructuring Preconditioners For Mixed Methods
with obvious modi cations for boundary cubes,
S
(1;j;k) 1
=
K
(1;j;k) 1
"
"
#
1 ?1 + 3h 2 ?1 1
S
=
K
S2(i;n?1;k) = K2(i;n?1;k) S
(i;j;1) 3
=
"
(i;1;k) 2
K
(i;j;1) 3
1
0 2a
2
#
"
1 ?1 + 3h 2 ?1 1
"
1 ?1 + 3h 2 ?1 1
#
"
"
1 ?1 + 3h 2 ?1 1
#
"
"
#
2a(1;j;k) 0 ; 0 0
" # 1 ?1 + 3h 0
S1(n?1;j;k) = K1(n?1;j;k) ?1 (i;1;k) 2
"
#
0
(n;j;k)
1
;
#
2a(i;1;k) 0 ; 0 0
#
0 0 ( 0 2a i;n;k) ;
#
0 0 2a(i;j;n) ;
2
i; k = 1; : : :; n;
#
2a(i;j;1) 0 ; 0 0
" # 1 ?1 + 3h 0
S3(i;j;n?1) = K3(i;j;n?1) ?1
j; k = 1; : : :; n;
i; j = 1; : : :; n:
The following statement plays very important role in all further arguments. It can be established by straightforward computations. Proposition 4.1. The matrix C^11 of the system (3.23) can be de ned by the relation (C^11u; v) = (4.2)
+
n nX ?1 X S2(i;j;k)u2(i;j;k) ; v2(i;j;k) + S1(i;j;k)u1(i;j;k); v1(i;j;k) +
n nX ?1 X
j;k=1 i=1 n nX ?1 X i;j =1 k=1
i;k=1 j =1
(i;j;k) (i;j;k) (i;j;k) S3 u3 ; v3
which is assumed to hold for any u; v 2 IRM , and ul ; vl are the restrictions of vectors u; v into IR2:
u
(i;j;k) 1
"
# (i;j;k) u = u(i+1;j;k) ;
u
(i;j;k) 2
"
# (i;j;k) u = u(i;j+1;k) ;
u
(i;j;k) 3
"
# (i;j;k) u = u(i;j;k+1) :
Now de ne on auxiliary cubic mesh T~C with vertices in the middle points of cubes C 2 TC . Enumerate the nodes of this mesh accordingly enumeration of the cubes of TC and introduce for each node (i; j; k) of T~C the matrices (i;j;k)
(4.3)
~(i;j;k)
Sl
~ (i;j;k)
= Kl
"
#
1 ?1 ?1 1 ;
l = 1; 2; 3:
16
S. Maliassov
Here coecients K~ l(i;j;k), l = 1; 2; 3 are given by ! h 3 h 1 a(i;j;k) + a(i;j?1;k) + a(i;j;k?1) + a(i;j?1;k?1) i ; (i;j;k) ~ K1 = min min min 2 ! 2 min h (i;j;k) (i?1;j;k) (i;j;k?1) (i?1;j;k?1) i K~ 2(i;j;k) = 32h 12 amin (4.4) ; + amin + amin + amin ! 3h 1 h (i;j;k) (i?1;j;k) (i;j?1;k) (i?1;j?1;k) i (i;j;k) ~ ; a + amin + amin + amin K3 = 2 2 min
n
o
(i;j;k) where amin = ; ; min a(i+ ;j+ ;k+ ) . =0;1 De ne the matrix C~ by the relation
n nX ?1 n nX ?1 X X S~2(i;j;k)u2(i;j;k) ; v2(i;j;k) + S~1(i;j;k)u1(i;j;k) ; v1(i;j;k) + i;k=1 j =1 j;k=1 i=1 n nX ?1 X S~3(i;j;k)u3(i;j;k) ; v3(i;j;k) +
(C~ u; v) = (4.5)
i;j =1 k=1
and consider the eigenvalue problem (4.6) C^11 u = C~ u; u 2 IRM : Proposition 4.2. The eigenvalues of the problem (4.6) belong to the interval [1=2; 1]. Proof. Consider rst eigenvalue problems (4.7) Sl(i;j;k)u = S~l(i;j;k)u; (S~l(i;j;k)u; u) 6= 0; u 2 IR2; l = 1; 2; 3; i; j; k = 1; : : :; n ? 1: Direct calculations show that eigenvalues of the problems (4.7) are = Kl(i;j;k) =K~ l(i;j;k). For l = 1 using (3.22) and (4.4) we can write ! (i;j;k) (i+1;j;k) 3h a a (i;j;k) K1 = 2 (i;j;k) (i+1;j;k) ; 2 ! ah + a i (i;j;k) (i;j ?1;k) (i;j;k?1) (i;j ?1;k?1) : + amin + amin + amin K~ 1(i;j;k) = 32h 12 amin
Suppose that a(i;j;k) = a(i+1;j;k) = a. Then K1(i;j;k) = 32h a and taking into account the ) are equal to Assumption A1 in the expression (4.4) of K~ 1(i;j;k) at least two terms a(min a. Thus, possible cases are either K~ 1(i;j;k) = 32h 12 4a; or K~ 1(i;j;k) = or K~ 1(i;j;k) =
3h 1
a + b);
2
2 (3
2
2 (2
3h 1
a + 2b);
where b a:
Substructuring Preconditioners For Mixed Methods
17
then we get, respectively,
either = a ; or = 2a ; or = a ; 2a 3a + b a+b with b a. So, we obtain 2 [1=2; 1]. ) = a and a(i+1;j;k) = b 6= a (suppose a b) then K (i;j;k) = 3h 2 ab , If a(i;j;k 1 2 a+b 3h (i;j;k) ~ K1 = 2 2b (by Assumption A1) and we obtain
(i;j;k) = K~ 1(i;j;k) = a +a b = 1 +1b=a 2 [1=2; 1]: K1 For l = 2; 3 we have the same arguments. Since all matrices Sl(i;j;k) and S~l(i;j;k), in the relations (4.2) and (4.5) are nonnegative then repeating the proof of Lemma 3.2 we obtain the conclusion of proposition. Now instead of the matrix (3.21) we de ne new matrix B by # 2" ~ 3 C + C12 C22?1C21 C12 + B B ?1 B B 12 22 21 12 7 B = 64 (4.8) C21 C22 5: B21 B22 Then we can formulate the following theorem Theorem 4.3. The matrix B de ned in (4.8) with the block C~ de ned in (4.5) is spectrally equivalent to the matrix A and cond(B ?1 A) 20: Proof. Proof is based on Proposition 4.2 and Theorem 3.6. We remind here that matrices B22 and C22 are diagonal and taking B as a preconditioner for the matrix A we have to develop procedure of solution the linear system of equations (4.9) C~ u = G; u 2 IRM : De ne on cubic mesh T~C standard partitioning into tetrahedra T~h. Direct calculations show that the matrix C~ de ned by (4.5) is nite element approximation of the boundary value problem ?r (~a ru) = g in ; (4.10) u=0 on @ ; on the partitioning T~h where function a~ is de ned to be constant on each cube C~ (i;j;k) 2 T~C : n (i+ ;j+ ;k+ ) o (4.11) ; x 2 C~ (i;j;k): a~(x) = ; ; min a =0;1
We have to stress that the function a~(x) is piecewise constant. Thus, any multilevel procedure which works well for such kind of problems (4.10) can be used. Below we outline the multilevel domain decomposition method (MGDD) [16, 17, 18, 19] which we used to solve the problem (4.9).
18
S. Maliassov
4.1. Multilevel domain decomposition method. Here we assume that provided the Assumption A1 we can choose a positive t 1 and for values l = 0; 1; : : :; t (l) (l)
nd grid domains ^ h as unions of pairwise disjoint cubes G i , i = 1; : : :; ml, with the edge length hl = 2?l H , where ml = 8l m and ht = h = 1=n. Then we partition each cube Gi(l), i = 1; : : :; m into tetrahedra in such a way that the resultant tetrahedral partitioning of the domain permits the application of the nite element method with piecewise-linear basis functions. Denote such tetrahedral partitions of the domain by
h(l) , l = 0; 1; : : :; t. Let us consider variational problem: for given g 2 L2 ( ) nd u 2 U = fv 2 H 1( ) : v = 0 on @ g such that (4.12)
Z
a~ ru rv d =
Z
gv d ;
8v 2 U;
where a~ is de ned by (4.11). Determine a sequence of spaces Uh(l) as a set of functions continuous in linear in eash tetrahedron from h(l) , l = 0; 1; : : :; t, and vanishing on @ and denote the dimensions of such spaces by Ml = dim Uh(l) . To approximate the problem (4.12) we consider the nite element problem: nd uh 2 Uh Uh(t) such that (4.13)
Z
a~ ruh rv d =
Z
gv d ;
8v 2 Uh;
which leads to the system (4.9) with the symmetric positive de nite M M matrix C~ and the vector G 2 IRM . Here M is equal to the dimension of the space Uh: M = Mt n3. Then we assume that the utilized tetrahedral partitioning of the domain is such that the system (4.9) is a classical 7-point di erence scheme. Following [16], de ne now the sequence of grids b h(l) as unions of faces @Gi(l?1) , i = 1; : : :; ml?1 with the edge length hl?1 b h(l) =
m[ l?1 i=1
@Gi(l?1) ;
and the sequence of grids h(l) as restrictions of grids h(l) into b h(l) , for l = 1; : : :; t. Also de ne the sequence of grids ?h(l) as unions of edges of cubes Gi(l), i = 1; : : :; ml, and the sequence of grids ?(hl?1=2) , which di er from the grids ?h(l?1) by additional nodes in the middle of the edges of Gi(l?1), i = 1; : : :; ml?1, for l = 1; : : :; t, as it is shown on Figure 3. Let us denote the set of the faces of the cubes Gi(l), i = 1; : : :; ml by Pi(l) , i = 1; : : :; sl, where sl is the number of these faces @Gi(l) , l = 0; 1; : : :; t. Then sl? b h(l) = S Pi(l?1) ; 1
(l)
?h =
i=1 sS l?1 i=1
@Pi(l?1) :
19
Substructuring Preconditioners For Mixed Methods
e
u
u
u e
u
e e u u e e e u e
e e
e
e
u
8 (a) Union of 8 cubes S Gi(l)
i=1
e
e
u
u e
e e
u e
u
u
e
u
u
e e u u e e e u
e e
u
(b) Fragment of the grid h(l)
e
e
e
u
u
u
(c) Fragment of the grid ?(hl?1=2)
u
u
e
e e
e
u
u
e e
e
u
u
u u
u
u
(d) Fragment of the grid ?h(l?1)
u
{ nodes of 1-st group
{ nodes of 3-rd group
e
{ nodes of 2-nd group
{ nodes of 4-th group
Figure 3. Fragments of the grids h(l) , h(l) , ?(hl?1=2) , ?h(l?1) and partitioning of the
nodes into groups
Thus, grid domains ^ h(l) consist of cubes, grid domains h(l) consist of tetrahedra, grids h(l) consist of squares, and grids ?h(l) and ?(hl?1=2) consist of edges of length hl , l = 0; 1; : : :; t. Let us partition the nodes of the grid domain h(l) into four groups (see Figure 3): 1. to the rst group we refer the vertices of the cubes Gi(l?1) (the nodes of ?h(l?1) ), 2. to the second group we refer the centers of the edges of these cubes (the nodes of ?(hl?1=2) which are not in ?h(l?1) ), 3. to the third group we refer the centers of the sides of the cubes (the nodes of h(l) which are not in ?(hl?1=2) ), 4. and to the forth group we refer all the remaining nodes which are at the same time the centers of the cubes Gi(l), i = 1; : : :; ml?1. According to such partitioning of the nodes any vector v 2 IRMl (Ml is the number of nodes of h(l) ) can be represented in the form v = (v1T ; v2T ; v3T ; v4T )T , where v1 2 IRMl? , v4 2 IRml? , and v3 2 IRsl? . Considering the equation (4.13) in the spaces Uh(l) we de ne 1
1
1
20
S. Maliassov
the sequence of matrices C~ (l) which according to the partitioning of nodes introduced above can be presented in the following block form 2 ~ (l) ~ (l) 3 C C 0 0 66 ~11(l) ~12(l) ~ (l) 77 C C C 0 6 7: ( l ) 21 22 23 C~ = 66 (4.14) (l) ~ (l) ~ (l) 7 ~ 4 0 C32 C33(l) C34(l) 75 0 0 C~43 C~44 Note that C~ii(l) , i = 1; 2; 3; 4 are diagonal matrices and the matrix C~ (t) coincides with the matrix C~ from (4.9). De ne Vh(l) and Wh(l?1=2) as the spaces of restrictions of functions from Uh(l) into h(l) and ?(hl?1=2) , respectively, l = 1; : : :; t. Then, following [17, 19] using relations l? ~ (l) mX hl Z a~ ru~ rv~ ds; 8u~ ; v~ 2 V (l) ; D u~ ; v~ = (4.15) h h h h h i=1 2 l 1
(h )
de ne symmetric positive de nite (Nl ? ml?1 ) (Nl ? ml?1 ){matrices 2 ~ (l) ~ (l) 3 D D 0 6 11 12 7 (4.16) D~ (l) = 64 D~ 21(l) D~ 22(l) D(l) 23 75 ; 0 D(l) 32 D(l) 33 and using the relations Z l? 2 c(l) sX du dv h (l?1=2) l a ~ h h ds; (4.17) 8 u ; D u; v = h ; vh 2 W h ds ds i=1 4 l? = 1
?(h
1 2)
de ne symmetric positive de nite nl nl {matrices
" c(l) (l) # ( l ) 12 ; c D = DD(l11) D (4.18) (l) 21 D 22 where nl = Ml ? ml?1 ? sl?1, for l = 1; : : :; t. Now de ne symmetric positive de nite Ml Ml matrices 2 ?1 0 66 D11 + DD12D22 D21 D + DD12D?1D D23 21 22 23 33 32 D(l) = 66 0 D32 D33 + D34 D44?1D43 4 0 0 D43 (4.19) where
= FlT [D11 D22 D33 D44] Fl;
c11 ? D12D22?1D21 ; D11 = D
D34 = C~34;
D44 = C~44 ;
0 0 D34 D44
3 77 77 = 5
Substructuring Preconditioners For Mixed Methods
2 0 66 D?I111D Fl = 66 220 21 D?I122D 4 33 32 0
0
0 0 I33 D44?1D43
0 0 0 I44
3 77 77 ; 5
21
and index l of matrices Dij , C~ij , and Iij , i; j = 1; 2; 3; 4 was skipped for simplicity. By making straightforward calculations [16] it can be shown that the following statement is true. (l) Lemma 4.4. D11 = 41 C~ (l?1) , l = 1; : : :; t. De ne now multilevel preconditioner for the matrix C~ from (4.9). Using (4.14){ (4.19) de ne the sequence of the preconditioners D(l) for matrices C~ (l) . Fix some integer c(1) ]?1 [D(1) ]?1. Then following [16] for l = 2; : : :; t de ne r 1 and de ne H11(2) = [D the sequence of Ml?1 Ml?1{matrices
3 2 r h i?1 Y I11(l) ? j H11(l)C~ (l?1) 5 C~ (l?1) ; (4.20) R11(l) = 4I11(l) ? j =1 i h 1 c11(l) = R11(l) ?1 ; D 4 where the parameters j , j = 1; : : :; r, are chosen such that the polynomial Yr Tr (x) = (1 ? j x) j =1
is least deviating from zero on the interval [d1; d2], where the constants d1; d2 are the boundaries of the spectrum of the matrix H11(l)D(l) 11. Then, de ne Ml Ml {matrices c(l) = FlT hD c11(l) D(l)22 D(l)33 D(l)44 i Fl; D (4.21) h c(l) i?1 H11(l+1) = D ; l = 2; : : :; t: Finally, set the matrix c(t) (4.22) D=D as an multilevel preconditioner for the matrix C~ of the problem (4.9). The following statements can be proved [16, 17].h i ?1 Lemma 4.5. The eigenvalues of the matrices D(l) C~ (l) belong to the interval p [1; b], where b = (7 + 19)=2, and h i?1 cond D(l) C~ (l) b < 5:68; l = 1; : : :; t: Lemma 4.6. The following estimates are valid:
p
if r = 3 then
cond D?1C~ 33?bp?b1 < 9:97;
if r = 4 then
cond D?1C~ max < 6:6;
22
S. Maliassov
where
q
p
p
max = 2 2b ? 2 b +p1 ? 3 b ? 2 : 4? b Let us apply the generalized conjugate gradient method to solve the system (4.9) with the matrix C~ : h i uk+1 = uk ? q1k D?1 k ? ek?1 (uk ? uk?1 ) (4.23)
?1
qk = kkDk k ?kC ? ek?1 ; 2
k 2
~
D 1
e0 = 0;
e k = qk
kk+1 k2D?1 kk k2D?1
k = 1; : : :; k"
with matrix D from (4.22) for the value r = 3 or r = 4, where k = C~ uk ? G and ~ )1=2, 2 IRM . k kC~ = (C; Choose the quantity k" in such a way that a given positive " (" 1) will surely satisfy the inequality
kuk +1 ? u kC~ " ku0 ? u kC~ ; where u = C~ ?1G, for any initial guess u0 2 IRM . (4.24)
"
Taking into account that the method (4.23) obeys the estimate k 2 q k ku ? u kC~ 1 + q2k ku0 ? u kC~ ; (4.25) p p where q = ( ? 1)=( + 1) and is an arbitrary but xed positive number such that cond D?1C~ , we can choose for the required value of k" the maximal integer satisfying the inequality k" lnln"=q2 : (4.26) The following statements can be established ([17]). Theorem 4.7. To solve the system (4.9) with the accuracy " in the sense of inequality (4.24) by the generalized conjugate gradient method (4.23) with the matrix D from (4.22) it is sucient to choose k" = [1:53 ln 2" ] in the case of r = 3 and k" = [1:22 ln 2" ] in the case of r = 4, where [z] denotes the integer part of number z. The number of arithmetic operations required in this case for the values r = 3 and r = 4 can be estimated from above by the quantities 75 M ln 2" and 70 M ln 2" , respectively. Note that the condition numbers of the matrices D?1C~ determined in the Lemma 4.6 do not depend on mesh size h and the jump of the coecients a~(x). So, applying
Substructuring Preconditioners For Mixed Methods
23
preconditioned conjugate gradient method to solve the problem (2.4) with the matrix B from (4.8) as a preconditioner for the matrix A and using multilevel domain decomposition method (MGDD) to solve the problem (4.9) with matrix C~ we establish the following results. Theorem 4.8. If we use MGDD method to solve problem (4.9) with matrix C~ then condition number cond(B ?1 A) does not depend on mesh size h and jump of the coecients a(x). Theorem 4.9. The number of operations for solving system
A = F by preconditioned conjugate gradient method with preconditioner B and with accuracy " in the sense
jjk +1 ? jjA "jj0 ? jjA ; is estimated by C N ln 2" , where = A?1 F, 0 2 IRN and C does not depend on N "
and jump of the coecients a(x).
5. Results of the numerical experiments. In this section the preconditioners
(3.21) and (4.8) are tested on the model problem
?r (a(x)ru) = f; u = 0;
in = [0; 1]3 on @
We present three numerical examples. In the rst example we use the preconditioner B~ in the form (3.21). The problem with M M {matrix C^11 is solved by preconditioned conjugate gradient with diagonal Jacoby preconditioner. In the second example we use the preconditioner B in the form (4.8). The problem with matrix C~ is solved by multilevel domain decomposition method as it is described in the section 4.1. The domain is divided into M = n3 cubes (n in each direction) and each cube is partitioned into 5 tetrahedra. The dimension of the original algebraic system is N = 10n3 ?6n2. The right hand side is generated randomly, and the accuracy parameter is taken as " = 10?6. The condition numbers of the preconditioned matrices B ?1A are calculated by the relation between the conjugate gradient and Lanczos algorithms [15]. The coecient a(x) is piecewise constant and is de ned to be (5.1)
(
(x; y; z) 2 [0:5; 1] [0:5; 1] [0:5; 1] a(x; y; z) = a; 1; elsewhere
The results are summarized in Tables 1 and 2, where niter and cond denote the iteration number and condition number, respectively. All experiments are carried out on Sun Workstation. Finally, the method of preconditioning described in this paper is used to solve the problem (1.2) with the constant right-hand-side function f (x) by the mixed nite
24
S. Maliassov
element method with the function a(x) in the following form (see Figure 4):
8 8 > > [0:2; 0:4] [0:2; 0:4] [0:2; 0:4][ > > > > [0:6; 0:8] [0:2; 0:4] [0:2; 0:4][ > > > > > > [0 :2; 0:4] [0:6; 0:8] [0:2; 0:4][ > > < > < a = 0:01; (x; y; z) 2 [0:6; 0:8] [0:6; 0:8] [0:2; 0:4][ > [0:2; 0:4] [0:2; 0:4] [0:6; 0:8][ (5.2) a(x; y; z) = > > > > [0:6; 0:8] [0:2; 0:4] [0:6; 0:8][ > > > > > > [0 :2; 0:4] [0:6; 0:8] [0:6; 0:8][ > > : > [0:6; 0:8] [0:6; 0:8] [0:6; 0:8] > : 1; elsewhere
9 > > > > > > > = : > > > > > > > ;
Again, the domain is the unit cube, the domain is divided into M = 403 = 64000 cubes. The dimension of the original algebraic system for the Lagrange multipliers (2.4) is N = 630400. Both preconditioners (3.21) and (4.8) are tested on this problem. With the preconditioner in the form (3.21), i.e. three-level preconditioner, it takes niter = 18 outer iterations to solve (2.4) with the accuracy " = 10?6. On each iteration the problem (3.20) is solved by preconditioned conjugate gradient method. It takes less than 40 iterations to solve (3.20) with the accuracy " = 10?8. With the preconditioner in the form (4.8), i.e. multilevel preconditioner when MGDD method is used to solve the problem (4.9), it takes niter = 22 outer iterations to solve (2.4). On each outer iteration it takes 18 iterations to solve (4.9) with the accuracy " = 10?8. In both cases it takes less then 12 minutes to obtain the resulting vectors q and u. The slices of the solution u by planes parallel to the xy{plane are shown in Figure 5. REFERENCES [1] T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second order elliptic problems, IMA Preprint #1172, 1993. (Submitted to Math. Comp.) [2] D.N. Arnold and F. Brezzi, Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates, RAIRO Model. Math. Anal. Numer., 19 (1985), 7{32. [3] O. Axelsson, On multigrid methods of the two-level type, Multigrid methods, Proceedings, KolnPorz, 1981, (Eds. W. Hackbush and U. Trottenberg), LNM 960, Springer, 1982, 352{367. [4] J.H. Bramble, J.E. Pasciak, and A.H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46 (1986), 361{369. [5] F. Brezzi, J. Douglas, Jr., R. Duran, and M. Fortin, Mixed nite elements for second order elliptic problems in three variables, Numer. Math., 51 (1987), 237{250. [6] F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed nite elements for second order elliptic problems, Numer. Math., 47 (1985), 217{235. [7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, Berlin, Heidelberg, 1991. [8] S.C. Brenner, A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular nite element method, SIAM J. Numer. Anal., 29 (1992), 647{678. [9] Z. Chen, Analysis of mixed methods using conforming and nonconforming nite element methods, RAIRO Math. Model. Numer. Anal., 27 (1993), 9{34.
25
Substructuring Preconditioners For Mixed Methods
Table 1. Solving C^11 with Jacoby preconditioning
20 20 20 N = 77600 a niter cond 1 15 6.20 10 17 7.59 100 18 7.86 1000 18 7.90 104 18 7.90 0:1 16 7.11 0:01 16 7.11 0:001 16 7.11 10?4 16 7.11
30 30 30 N = 264600 niter cond 15 6.15 17 7.59 18 7.85 18 7.90 18 7.90 16 7.10 16 7.10 16 7.10 16 7.10
40 40 40 N = 630400 niter cond 14 6.03 17 7.59 17 7.84 17 7.88 17 7.88 16 7.10 16 7.10 16 7.10 16 7.10
50 50 50 N = 1235000 niter cond 14 6.02 17 7.58 17 7.83 17 7.88 17 7.88 16 7.10 16 7.10 16 7.10 16 7.10
Table 2. Solving C~ by MGDD method
20 20 20 N = 77600 a niter cond 1 20 10.6 10 21 12.6 100 22 12.8 1000 22 12.9 4 10 22 12.9 0:1 21 11.7 0:01 22 11.8 0:001 22 11.9 ? 4 10 22 11.9
30 30 30 N = 264600 niter cond 19 10.4 21 12.8 22 12.9 22 12.9 22 12.9 22 11.8 22 11.8 22 11.8 22 11.8
40 40 40 N = 630400 niter cond 19 10.3 22 12.9 22 12.9 22 12.9 22 12.9 21 11.9 21 11.9 21 11.9 21 11.9
50 50 50 N = 1235000 niter cond 19 10.3 22 12.9 22 13.0 22 13.0 22 13.0 21 11.9 21 11.9 21 12.0 21 12.0
[10] Z. Chen, R.E. Ewing, Yu. Kuznetsov, R. Lazarov, and S. Maliassov, Multilevel preconditioners for mixed methods for second order elliptic problems, IMA Preprint #1269, 1994. (Submitted to SIAM Numer. Anal.) [11] J.Douglas and J. Wang, A new family of mixed nite element spaces over rectangles, Comp. Appl. Math., 12 (1993), 183{197. [12] R.E. Ewing, Yu. Kuznetsov, R. Lazarov, and S. Maliassov, Substructuring preconditioning for nite element approximations of second order elliptic problems. I. Nonconforming linear elements for the Poisson equation in parallelepiped, ISC-94-2-MATH Technical Report, TAMU, 1994. [13] R.E. Ewing, Yu. Kuznetsov, R. Lazarov, and S. Maliassov, Preconditioning of nonconforming nite element approximations of second order elliptic problems, Proceedings of the Third Int. Conf. on Advances in Numerical Methods and Applications, Bulgaria, 1994 (I.T. Dimov, Bl. Sendov, P.S. Vassilevski, eds.), 101{110 [14] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, Massachusetts, 1985. [15] G.M. Golub and C.F. Van Loan, Matrix Computations, Johns Hopkins University Press, 1989. [16] Yu. Kuznetsov, Multigrid domain decomposition methods for elliptic problems, Proceedings of
26
S. Maliassov
Figure 4. Function a(x) for the model problem (5.2)
20
200
15
150
10
100
5
50
0
0
(a) Slice on z = 0:1
(b) Slice on z = 0:3
200
50 45 40
150
35 30
100
25 20 15
50
10 5 0
0
(c) Slice on z = 0:5
(d) Slice on z = 0:7
Figure 5. Slices of the solution parallel to xy -plane
Substructuring Preconditioners For Mixed Methods [17] [18] [19] [20] [21] [22]
27
VII-th Int. Conf. on Comput. Methods for Applied Science and Engineering, 2 (1987), 605{ 616. Yu. Kuznetsov, Multigrid domain decomposition methods, Proceedings of 3rd International Symposium on Domain Decomposition Methods, 1989 (T. Chan, R. Glowinski, J. Periaux, O. Widlund, eds.), SIAM, Philadelphia, 1990, 290{313. Yu. Kuznetsov, Multilevel domain decomposition methods, Appl. Numer. Math., 6 (1989/90), 186{193. S. Maliassov, Numerical analysis of the three-body problem with Coulomb interaction, Dissertation, Institute of Numerical Mathematics, Moscow, Russia, 1993 (in Russian). L. D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart{Thomas mixed method, SIAM J. Numer. Anal., 22 (1985), 493{496. J. C. Nedelec, Mixed nite elements in R3 , Numer. Math., 35 (1980), 315{341. P.A. Raviart and J.M. Thomas, A mixed nite element method for second order elliptic problems, In: Mathematical aspects of the FEM, Lecture Notes in Mathematics, 606, Springer-Verlag, Berlin & New York (1977), 292{315.
