Z. CHEN ET AL. []). This space is nite dimensional and dened locally on each element T T h, so let V h (T )=V h j T and W h (T )=W h j T. Then the mix

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1 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS FOR SECOND ORDER ELLIPTIC PROBLEMS ZHANGXIN CHEN, RICHARD E. EWING, YURI A. KUZNETSOV, RAYTCHO D. LAZAROV, AND SERGUEI MALIASSOV Abstract. A new approach of constructing algebraic multilevel preconditioners for mixed nite element methods for second order elliptic problems with tensor coecients on general geometry is proposed. The linear system arising from the mixed methods is rst algebraically condensed to a symmetric, positive denite 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 classications. N, N, F. Introduction. Let be a bounded domain in IR d, d = or, with the polygonal We consider the elliptic problem (:) ;r (aru) =f in u = where a(x) is a uniformly positive denite, bounded, symmetric tensor and f(x) L () (H k () = W k () is the Sobolev space of k dierentiable functions in L ()). Let ( ) S denote the L (S) inner product (we omits if S = ), and let V = H(div)= v ; L () d : rv L () W = L (): Then (.) is formulated in the following mixed form for the pair (u) V W : (:) (rw)=(fw) 8w W (a ; v) ; (u rv) = 8v V: It can be easily seen that (.) is equivalent to (.) through the relation = ;aru: To dene a nite element method, we need a partition T h of into elements T, say, simplexes, rectangular parallelepipeds, and/or prisms. In T h,we also need that adjacent elements completely share their common edge or face h denote the set of all interior edges (d = ) or faces (d =)e of T h. Let V h W h V W denote some standard mixed nite element space for second order elliptic problems dened over T h (see, e.g., [8], [9], [], [], [], [9], [], and Z. Chen, R. Ewing, R. Lazarov, S. Maliassov, Institute for Scientic Computation and Department of Mathematics, Texas A&M University, Teague Research Center, College Station, TX 778{. Y. Kuznetsov, Institute of Numerical Mathematics, Russian Academy of Sciences, -a Leninski Prospect, 7 Moscow, Russia. R. Lazarov, Institute of Mathematics, Bulgarian Academy of Sciences, Soa, Bulgaria.

2 Z. CHEN ET AL. []). This space is nite dimensional and dened locally on each element T T h, so let V h (T )=V h j T and W h (T )=W h j T. Then the mixed nite element method for (.) is to nd ( h u h ) V h W h : (:) (r h w)=(fw) 8w W h (a ; h v) ; (u h rv) = 8v V h : It has been known that, due to its saddle point property, itisdicult to solve the linear system arising from (.). In particular, it is hard to construct ecient preconditioners for the mixed method system. While some preconditioning algorithms for solving the saddle point system have been proposed and studied (see, e.g., [], [], [7], [], [], []), their eciency strongly depends on the geometry of the domain, the coecient matrix a(x), and the type of nite elements exploited. Especially, the preconditioners in these papers can be very expensive toobtain. An alternate approach was suggested by means of a nonmixed formulation. Namely, itisshown that the mixed nite element method is equivalent to a modication of the nonconforming Galerkin method [], [], [], [8]. The nonconforming method yields a symmetric and positive denite problem (i.e., a minimization problem). This is explained below and in the next section in detail. The constraint V h V says that the normal components of the members of V h are continuous across the interior boundaries h. Following [], we relax this constraint onv h by dening ~V h = fv L () : vj T V h (T ) for each T T h g: We then need to introduce Lagrange multipliers to enforce the required continuity on ~V h, so dene L h = L [ e@t h e : j e V h j e for each h 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 [] we introduce the projection of the coecient, i.e., h = P h a ;, where P h is the L -projection onto W h. Then the hybrid form of the mixed method for (.) is to nd ( h u h h ) Vh ~ W h L h such that P T T h (r h w) T =(fw) 8w W h (:) P ( h h v) ; T T (uh h rv) T ; ( h v T = 8v Vh ~ P T T h ( h T = 8 L h : Note that the last equation in (.) enforces the continuity requirement mentioned above, so in fact h V h. In [], [], [], and [8], it was shown that the solution to (.) can be obtained from a certain modied nonconforming Galerkin method by means of augmenting the latter with bubble functions. In this paper, following [] and [], we show that the linear system generated by (.) can be algebraically condensed to a symmetric, positive denite 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.

3 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS The main objective ofthis 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 h and u h are recovered via these multipliers. The construction of multilevel preconditioners for the mixed methods is inspired by the fundamental work [7], [], 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 [], [] and their multigrid versions were described in [], [] for model elliptic equations. In addition, the superelement approach used here to estimate condition numbers for two level methods is based on that used in [], [], [8], [], and [7]. This paper unies and renes the results previously announced in [8] and [9]. Briey, the approach used to construct preconditioners includes two main stages. First, using the idea of partitioning (decomposing) a parallelepiped grid into tetrahedral substructures a two-level preconditioner is constructed for a block \7-point" algebraic system with blocks on the coarse level, and the condition number of the preconditioned matrix is estimated. A detailed description of procedures to construct such preconditioners for parallelepiped grids can be found in [8], [], [], and []. The explicit bounds of spectrum of the preconditioned matrix are obtained with help of the superelement approach [8], []. On the second stage, introducing a special rotation we reduce the above block 7-point algebraic system to a series of plane problems and an exact 7-point-scheme problem with one unknown per parallelepiped. The constructed preconditioners are spectrally equivalent to the original stiness matrix and their arithmetic cost depends on the method of solving the 7-point problem on the coarsest level. In our implementation we use a method of separation of variables although some variants of multigrid methods or domain decomposition techniques can be used. 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. The case where a(x) is a diagonal tensor and is a regular parallelepiped is rst analyzed in detail for the multilevel preconditioners. Then the analysis is extended to the case in which a(x) is a full tensor and is a rather general domain. Throughout this paper, only the three-dimensional problem is discussed the two-dimensional case is treated as a special case of the present technique. The rest of the paper is organized as follows. In the next section we consider an elimination procedure for (.). Then, in x wedevelop multilevel preconditioners for the resulting linear system. In x, we consider arbitrary tetrahedral meshes and full tensors. Finally, inx extensive numerical results are presented for both regular and logical parallelepipeds to illustrate the present theory.. The mixed nite element method. We now consider the most useful partition T h of into tetrahedra. The lowest-order Raviart-Thomas-Nedelec space [], [9] dened over T T h is given by V h (T )= ; P (T ) ; (x y z)p (T ) W h (T )=P (T ) L h (e)=p (e)

4 Z. CHEN ET AL. where P i (T ) is the restriction of the set of all polynomials of total degree not bigger than i tothesett T h. For each T in T h,let f T = jt j (f) T where jt j denotes the volume of T. Also, set h =( ij )and h j T =( T T T )= (q T + t T x q T + t T y q T + t T z). Then, by the rst equation of (.), it follows that (:) t T = f T =: Now, take v = ( ) in T and v = elsewhere, v = ( ) in T and v = elsewhere, and v = ( ) in T and v = elsewhere, respectively, in the second equation of (.) to obtain (:) ( P i= ji Ti ) T + P i= jei T ji(j) T hj e i T = j = where je i T j is the area of the face ei T, and i T =(i() T (( ij ) T ) ;. Then (.) can be solved for qt j : (:) qt j = ;P i= jei T j(t j i() T + T j i() T i() T i() + T j i() T ) h j e i T ; f T (P i= T ji ( ix + i y + i z) ) T j = : T ). Let T =( T ij )= Let the basis in L h be chosen as usual. Namely, take =onone face and = elsewhere in the last equation of (.). Then, apply (.) and (.) to see that the contributions of the tetrahedron T to the stiness matrix and the right-hand side are m T ij = i T T j T F T i = ; (Jf T i T ) T jt j +(J f T i T ) e i T T T h where i T = jei T ji T and Jf T = f T (x y z)=. Hence we obtain the system for h : (:) M = F: After the computation of h, we can recover h via (.) and (.). Also, if u h is required, it follows from the second equation of (.) that! u T = X ( h (x y z)) T + h j jt j e i T ((x y z)t) i e i T T T h : i= The above result is summarized in the following lemma. Lemma. Let M h ( )= P T T h ( T T ( T F h () =; P T T h jt j (Jf ) T ( T + P T T h (J f T L h L h where J f is such that J f j T = J f T. Then h L h satises (:) M h ( h )=F h () 8 L h

5 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS where L h = f L h : j e =foreach Note that there are at most seven nonzero entries per row in the stiness matrix M. Also, it is easy to see that the matrix M is a symmetric and positive denite matrix moreover, if the angles of every T in T h are not bigger than =, then it is an M-matrix. Finally, (.) corresponds to the P nonconforming nite element method system, as described below. This equivalence is used to construct our multilevel preconditioners later. Let (:) N h = fv L () : vj T P (T ) 8T T h v is continuous at the barycenters of interior faces and vanishes at the barycenters of faces Proposition. Let f h = P h f. Then (.) corresponds to the linear system produced by the problem: Find h N h such that (:7) a h ( h ')=(f h ') 8' N h where a h ( h ')= P T T h ( ; h r h r') T. Proof. From the denition of the basis f h i g of N h, for each T T h wehave h i j T = jt j i T ((x y z) ; p l ) i = l for some barycenter p l. Then, we see that ( ; h r i h r j h ) T = i T T j T whichism T ij. Also, note that for any linear functions and on a tetrahedron T T h (:8) ( ) T = jt jp i= (p i)(p i ) where the p i 's are the barycenters of the faces of T,sothat F T i = ; (Jf T i T )T jt j +(J f T i T ) e i T which is(f h h i ) T. = ; f T ( h i ) T + jt jf T je i T j ( h i ) e i T = f T ; h i T. Multilevel preconditioners over a cube. In this section we consider multilevel preconditioners for (.) based on partitioning regular parallelepipeds into tetrahedral substructures, following the ideas in [8] and [9]. Here we treat the case where is a unit cube and a(x) is a diagonal tensor. A general case is considered in the next section.

6 Z. CHEN ET AL... Two level preconditioners. Let C h = fc (ijk) g be a partition of into uniform cubes with the length h = =n, where (x i y j z k ) is the right back upper corner of the cube C (ijk). Next, each cubec (ijk) is divided into two prisms P = P (ijk) and P = P (ijk) as shown in Figure. The resulting partition of is denoted by P h. Finally, we divide each prism into three tetrahedra as illustrated in Figure and denote this partition of into tetrahedra by T h. Let W ch be the space of piecewise constants associated with C h, and P ch be the L -projection onto W ch. To dene our preconditioners, we introduce h = P ch a ; in the hybrid form (.) instead of h = P h a ;. Obviously, Lemma and Proposition are still valid for this modication since T h is a renement ofc h. With this modication, ; h is a constant oneach cube. For notational convenience, we drop the subscript h and simply write ; h =diag(a a a ). Let N h be the nonconforming nite element space associated with T h as dened in (.), and let its dimension be N. All the unknowns on the faces are excluded. For any function v h N h, we denote by v IR N the corresponding vector of its degrees of freedom. Introduce the inner product X (:) (u v) N = h u h (p i )v h (p i ) u h v h N h p i@t h where the p i 's are the barycenters of the interior faces. By (.8) the norm induced by (.) is equivalent tothel -norm on. For each prism P = P (ijk) P h,denoteby Nh P the subspace of the restriction of the functions in N h onto P. For each v Nh P,we indicate by v P its corresponding vector. The dimension of Nh P is denoted by N P. Obviously, for a prism without faces its dimension is, i.e., N P =. The local stiness matrix M P on prism P P h is given by (:) (M P u P v P ) N P = X T P ( ; h ru h rv h ) T : Then the global stiness matrix is determined by assembling the local stiness matrices: (:) (Muv) N = X P P h (M P u P v P ) N P : Now we consider a prism P of a cube that has no face on the and enumerate the faces s j, j = ::: of the tetrahedra in this prism as shown in Figure. Then the local stiness matrix of this prism has the following form: M P = h a ;a a ;a a ;a a ;a a ;a a ;a ;a a + a ;a ;a a + a ;a ;a ;a ;a (a + a) ;a ;a ;a ;a ;a (a + a) 7

7 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS 7 ; ;; ; ;; ; ;; ; ;; ; ;; ; ;; ; ;; ; ;; ; ;; Figure. The partition of a cube into prisms and tetrahedra. which we write as (:) M P = h M M : M M Along with matrix M P wealsointroduce the matrix B P dened on the same space Nh P : (:) B P = h where B = with some parameter b to be specied later. Proposition. It holds that M M M B a + a + b ;b ;a ;b a + a + b ;a ;a a + a ;a a + a kerm P =kerb P : Proof. It is easy to see from the denitions of M P and B P that 7 kerm P =kerb P = fv =(v v v ) T IR : v i = v i= g:

8 8 Z. CHEN ET AL. z,,,,,,,,, 8 7 x y (a) P s =( ) s =( ) s =( ) s 7 =( ) s 9 =() s =( ) s =( ) s =( ) s 8 =( ) s =(),,,,,, z x,,, y 8 7,,,,,,,,, (b) P s =( ) s =( ) s =( ) s 7 =( ) s 9 =() s =( ) s =( ) s =( ) s 8 =( ) s =() Figure. A local enumeration of faces in a prism. Remark. If the prism P P h has a face then the matrix M P does not have the rows and columns which correspond to the nodes on that face, and the modication of B is obvious.

9 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS 9 We now dene the N N matrix B by the following equality: (Buv) N = X P P h (B P u P v P ) N P 8u v IR N : Since the matrix B is used for preconditioning the original problem (.), it is important to estimate the condition number of B ; M: (:) Mu = Bu: Lemma. Let P satisfy the equality (:7) M P u P = P B P u P P P h : Then we have (Muu) N (:8) max (Buu) N = (Buu) N (Muu) N max P and min P P h (Buu) N = (Buu) N Proof. For each P P h, it follows from (.7) that (M P u P u P ) N P = P (B P u P u P ) N P : min P P h P : It then follows from the fact that all local stiness matrices are nonnegative that X X (M P u P u P ) N P = P P h P P h P (B P u P u P ) N P max P P h P Hence from the denitions of M and B, we see that X (Muu) N max P P h P (Buu) N : P P h (B P u P u P ) N P : Consequently, the rst inequality in (.8) is true. The same argument can be used to show the second inequality. From Lemma, we see that, to estimate the condition number of B ; M, it suces to consider the local problem (.7). Using a superelement analysis [], to estimate max P and min P, it suces to treat the worst case where the prism P P h P P h P P h has no face on the From (.) and (.), a direct calculation shows that the eigenvalue P is within the interval [ ; P + P ], where s + a + a + a ; a a b (:9) P =! a =b : ( + a =a + a =a + a =b) We now consider two useful choices of b. The below isgiven by a a max : P P h a a Theorem. The eigenvalues of problem (.) with the parameter b = a belong to the interval h p p i + ;

10 Z. CHEN ET AL. and the condition number is thus estimated by cond(b ; M) +8 + : If the parameter b is chosen by b ; = a ; +a; (.) are within the interval " ( + ) ; +a;, then the eigenvalues of problem r! r!# ( + ) and the condition number is then estimated as follows: cond(b ; M) +8: Proof. We only prove the case where b = a the other choice can be shown with the same argument. When b = a,the P can be written as P = Then we consider the + a a + a a s + a a + a p x x ; x : ; A : a f (x)= Note that f + is a nondecreasing function and f ; is a nonincreasing function. Hence the desired result follows from the denition of. We stress that the condition number of the matrix B ; M is bounded by aconstant independent of the step size of the mesh h. Furthermore, the second choice of b has a much better estimate of the condition number than the rst choice. Hence only the second choice will be considered in the following sections. Since we introduced a two level subdivision, the matrix B can be referred to as a two level preconditioner... Three level preconditioners. While the preconditioner B has good properties, it is not economical to invert it. In this section we propose a modication of the matrix B and consider its properties and computational scheme. Toward that end, we divide all unknowns in the system into two groups:. The rst group consists of all unknowns corresponding to faces of the prisms in the partition P h, excluding the faces (see Figure ).. The second group consists of the unknowns corresponding to the faces of the tetrahedra that are internal for each prism (these are faces s 9 and s on Figure ). This splitting of the space IR N induces the presentation of the vectors: v T = (v v ), where v IR N and v IR N. Obviously, N = N ; n. Then the matrix B can be presented in the following block form: B B (:) B = dimb B B = N : Denote now by ^B = B ; B B ; B the Schur complementofb obtained by elimination of the vector v. Then B = ^B + B B ; B, so the matrix B has the form (:) B = ^B + B B ; B B B B :

11 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS Note that for each prism P P h the unknowns on the faces s 9 and s (see Figure ) are connected only with the unknowns associated with this prism and therefore can be eliminated locally that is, the matrix B is block diagonal with blocks and can be inverted locally (prism by prism). Thus matrix ^B is easily computable. The proposed modication of the matrix B in (.) is of the form ~B = where ~ B is to be dened later. ~B + B B ; B B B B... Group partitioning of grid points. For the sake of simplicity of representation of matrices and computational schemes we introduce the partitioning of all nodes h into the following three groups. Denote by s (ijk) rlm the face of the cube C (ijk) with vertices rlm (see Figure ). z x y 7 8 ; ;; ; ;; ; ;; Figure. A cube C (ijk).. First, we group the nodes on the faces s (ijk) and s (ijk) 7 i j k = n we denote the unknowns at these nodes by VI (ijk) `, ` =, i j k = n.. Second, we take the nodes on the faces perpendicular to the x, y, andz axes: (i) s (ijk) s(ijk) i = n j k = n we denote the unknowns at these nodes by Vx (ijk) `, ` =, i = n, j k = n. (ii) s (ijk) s(ijk) 7 j = n i k = n we denote the unknowns at these nodes by Vy (ijk) `, ` =, j = n, i k = n. (iii) s (ijk) s(ijk) i j = n k = n we denote the unknowns at these nodes by Vz (ijk) `, ` =, i j = n, k = n.

12 Z. CHEN ET AL.. Finally, we take the remaining nodes on the faces s (ijk) s(ijk) s(ijk) s(ijk) 8 i j k = n we denote the unknowns at these nodes by VA (ijk) `, ` =, i j k = n.... Three level preconditioners. We partition each cubec (ijk) into the left and right prisms P p (ijk), p = see Figure. Below we skipthe indices `(i j k)' and the superscript `P 'whennoambiguity occurs. In the local numeration (Figure ) the matrices B and B corresponding to the left and right prisms have the form (.). We rewrite these matrices in the above group partitioning: B = h a + a + b ;b ;a ;a ;b a + a + b ;a ;a ;a a a ;a a ;a ;a a a ;a a ;a ;a ;a ;a a + a ;a ;a ;a a + a 7 B = h a + a + b ;b ;a ;a ;b a + a + b ;a ;a a ;a ;a a ;a a a ;a a ;a a ;a ;a ;a ;a a + a ;a ;a ;a a + a 7 : The partitioning of nodes into the above three groups induces the following block forms of the matrices B p, p =: Bp B (:) B p = p p = B p B p where the blocks B p correspond to the unknowns of the last group and the blocks B p correspond to the unknowns of the rst and second groups. We eliminate the unknowns of the last group from eachmatrixb p, p =, which is done locally on each prism. Then we get the matrices ^B p = B p ; B p B ; p B p p =

13 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS where B B ; B = h a a a +a a a a +a a +a a a +a a a +a a a +a a a a a +a a a a +a a +a a a a +a a a +a a a +a a a a +a a a a a a +a a +a a a a +a a a +a a a +a 7 and a similar form holds for B B ; B. Following [8], we introduce on each prism the following modication of the matrices ^Bp : B = h a + a + b + s ;b ;a ;a ;s= ;s= ;b a + a + b + s ;a ;a ;s= ;s= ;a a ;a a ;a a ;a a s + s ;s= ;s= s + s ;s= ;s= with some parameters s and s. Proposition. The matrices ^Bp, p =, and B have the same kernel, i.e., 7 ker ^Bp =kerb : Proof. It can be easily checked that ker ^Bp =kerb = fv =(v v v 8 ) T IR 8 : v i = v i= 8g p = : We now consider the eigenvalue problem (:) ^Bp u = B u u IR 8 p = with the following choices of s and s. Proposition 7. For the case where s i =a i a =(a i + a ), i =, the eigenvalues of problem (.) belong to the interval + + ( ; p ) + + ( + p ) : If we choose s i = maxfa i a g, i =, the eigenvalues of problem (.) are within the interval + + ( ; p ) + + ( + p ) :

14 Z. CHEN ET AL. Both cases have the same estimate of the condition number cond(b ; ^B p ) + p : Proof. A direct calculation shows that [ ; + ]where ; =min ( a i + a + a a i +a a i s i ; s! ; a=(a is i )+a =s i : i = +a =a i +a =s i ) and + =max ( a i + a + a a i +a a i s i + s! ; a=(a is i )+a =s i : i = +a =a i +a =s i ) : With s i = a i a =(a i + a ), i =, and the denition of, it can be seen as in Theorem that r! ; + + ; ; +=+ = + and r! ; +=+ = : + Note that ; +=+ = + so that the rst case follows. The same argument applies to the second case. Now we dene a new matrix on each prism: (:) Bp ~ B + B = p B p ; B p B p p =: B p B p As remarked before, in the case where a cube C has nonempty intersection the matrices B, B p, and B p, p =, do not have the rows and columns corresponding to the nodes on the boundary. For each prismp P h wenow consider the eigenvalue problem: (:) B P u = ~ B P u where B P = Bp P is dened in (.) and B ~ P = B ~ P p in (.), p =. Below we only consider the simpler choice: s i =maxfa i a g, i =. Proposition 8. The eigenvalues of problem (.) belong to the interval + + ( ; p ) + + ( + p ) : Moreover, on each prism P P h the eigenvalues of the problem (:) M P u = ~ B P u

15 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS are within the interval [ ; + ], where =(+) r +! + p : + Proof. The rst statement follows directly from Proposition 7, and the second one then follows from Theorem. Now we dene the symmetric positive-denite N N matrix ~ B by ( ~ B u v )= X P P h (B u P v P ) where v u IR N, and u P and v P are their respective restrictions on the prism P. As in (.), we introduce the matrix ~B (:7) B ~ + B = B ; B B : B B Using Proposition 8 and the same proof as in Theorem, we have the following theorem. Theorem 9. The matrix ~ B dened in (.7) is spectrally equivalent to the matrix M, i.e., Moreover, ~ B M ~ B: (:8) cond( ~ B ; M) = ( + 8k)( + p ): Instead of the matrix B in the form (.) we take the matrix ~ B in (.7) as a preconditioner for the matrix M. Because we have introduced a two-level subdivision of the matrix ~ B, the matrix ~ B can be considered as a three-level preconditioner. As we noted earlier, the matrix B is block-diagonal and can be inverted locally on prisms. So we concentrate on the linear system (:9) ~ B u = G: In terms of the group partitioning in x.., the matrix ~ B has the block form (:) ~ B = C C C C where the block C corresponds to the nodes from the second group, which are on the faces of tetrahedra perpendicular to the coordinate axes. From the denition of B, it can be seen that the matrix C is diagonal. In the above partitioning, we present u and G in (.9) in the form u G (:) u = G = : u G Then, after elimination of the second group of unknowns: u = C ; (G ; C u )

16 Z. CHEN ET AL. we get the system of linear equations (:) (C ; C C ; C )u = G ; C C ; G ~ G where the vector u and the block C correspond to the unknowns from the rst group, which have only two unknowns per each cube. The dimension of vectors u and G is equal to (:) dim(u )=n : The above simplication of (.9) is carried out in detail in the next section.... Computational scheme. We now consider the computational scheme for (.9). In terms of the unknowns introduced in x..: ui (ijk) ` GI (ijk) ` ` = i j k = n ux (ijk) ` Gx (ijk) ` ` = i = n j k = n uy (ijk) ` Gy (ijk) ` ` = j = n i k = n uz (ijk) ` Gz (ijk) ` ` = k = n i j = n the system (.9) with a(x) can be written as ; ui ; (ijk) ; ( ; i )ux (i;jk) ; ( ; in )ux (ijk) (:) ;( ; j )uy (ij;k) ; ( ; jn )uy (ijk) ;( ; k ) uz (ijk;) ; ( ; kn ) uz (ijk) = h GI (ijk) i j k = n and ux (ijk) ; ui (i+jk) ; ui (ijk) = h Gx(ijk) i = n; jk= n (:) uy (ijk) ; ui (ij+k) ; ui (ijk) = uz (ijk) ; ui (ijk+) ; h Gy(ijk) j = n; ik= n ui (ijk) = h Gz(ijk) k = n; i j = n where ij (the Kronecker symbol) is introduced to take into account the Dirichlet boundary conditions. Eliminating unknowns ux (ijk) `, uy (ijk) `, uz (ijk) `, ` =, from equations (.), we obtain the block \seven-point" scheme with -blocks for

17 (:) MULTILEVEL PRECONDITIONERS FOR MIXED METHODS 7 the unknowns ui (ijk) `, ` =, i j k = n. From (.) we have ux (ijk) = Gx (ijk) + h ui (i+jk) + ui (ijk) uy (ijk) = uz (ijk) = + i = n; j k = n Gy (ijk) + h ui(ij+k) + ui(ijk) Gz (ijk) + h j = n; i k = n ui (ijk+) ui (ijk) k = n; i j = n: Substituting (.) into (.), we see that ; ui (ijk) ; ( ; ; i ) ui (i;jk) + ui (ijk) ;( ; in ) ui (i+jk) + ui (ijk) ;( ; j ) ui (ij;k) + ui (ijk) (:7) ;( ; jn ) ui (ij+k) + ui (ijk) ;( ; k ) ui (ijk;) + ui (ijk) ;( ; kn ) ui (ijk+) + ui (ijk) = g (ijk) i j k = n where (:8) g (ijk) = h GI (ijk) +(; i ) Gx(i;jk) +(; in ) Gx(ijk) +( ; j ) Gy(ij;k) +(; jn ) Gy(ijk) +( ; k ) Gz (ijk;) +( ; kn ) Gz (ijk) : To solve system (.7) we introduce the rotation matrix Q = p ;

18 8 Z. CHEN ET AL. and the new vectors v (ijk) =(v (ijk) v (ijk) ) T, i j k = n, given by (:9) v (ijk) = Q ui (ijk) i j k = n: Then multiplying both sides of the matrix equation (.7) by the matrix Q and using the relation (:) ui (ijk) = Q T v (ijk) i j k = n we obtain the following problem for the unknowns v (ijk) : (:) = ; v (ijk) ; ( ; i ) v (i;jk) + v (ijk) ; ;( ; in ) v (i+jk) + v (ijk) ; ;( ; j ) v (ij;k) + v (ijk) ; ;( ; jn ) v (ij+k) + v (ijk) ;v ;( ; k ) (ijk; )+v (ijk) ;( ; kn ) ;v (ijk+) + v (ijk) = Q g (ijk) ~g (ijk) i j k = n: It is easy to see that problem (.) can be decomposed into the following two independent problems: (:) v (ijk) ; ( ; i ) v (i;jk) + v (ijk) ;( ; in ) ;( ; j ) ;( ; jn ) ;( ; k ) ;( ; kn ) v (i+jk) v (ij;k) + v (ijk) + v (ijk) v (ij+k) + v (ijk) v (ijk;) + v (ijk) v (ijk+) + v (ijk) i j k = n =~g (ijk)

19 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS 9 and (:) v(ijk) ; ( ; i ) v (i;jk) + v (ijk) ;( ; in ) ;( ; j ) ;( ; jn ) v (i+jk) + v (ijk) v (ij;k) + v (ijk) v (ij+k) + v (ijk) =~g (ijk) i j = n 8 k = n: That is, we reduced the linear system (.) of dimension (n ) to one linear system of equations (.) of dimension n and n linear systems of equations (.) of dimension n. For all these problems the method of separation of variables can be used, as shown in (.) and (.8) below. After we nd the solution of these problems we easily retrievevectors ui (ijk) by using the relations (.).... A method of separation of variables. In this section we consider a method of separation of variables for solving problems (.) and (.). Problem (.) can be represented in the form (:) C () v =~g v ~g IR n where C () = C I I + I C I + I I C I is the (n n)-identity matrix, denotes the tensor product of matrices, and C takes the form (:) C = If C is factorized by ; ; ; C = Q Q T ; ; ; where is an (n n)-diagonal matrix and Q is an (n n)-orthogonal matrix (Q ; = Q T ), then the matrix C () can be rewritten as follows: C () = Q () () Q () 7 : where Q () = Q Q Q () = I I + I I + I I :

20 Z. CHEN ET AL. Note that Q () is an (n n )-orthogonal matrix and () is an (n n )-diagonal matrix. We can now use the following method to solve the system (.): () ~ f = ; Q () T ~g (:) () () w = ~ f () v = Q () w: The same argument can be exploited to solve (.). rewritten as (:7) C () v =~g v ~g IR n where and the (n n)-matrix K is given by Again, if we write K as K = C () = K I + I K 9 ; ; ; K = R D R T ; ; ; 9 7 : Problem (.) can be where D is an (n n)-diagonal matrix and R is an (n n)-orthogonal matrix, we can rewrite the matrix C () as follows: C () = Q () () Q ()T where Q () = R R and () = D I + I D. Then the system (.) can be solved with the following method: () ~ f = ; Q () T ~g (:8) () () w = ~ f () v = Q () w:... Preconditioned conjugate gradient method. Wenowsolvesystem (.) by a three-step preconditioned conjugate gradient method in the following form: Given ;, IR N,nd k, k = K,suchthat (:9) (k+) = k ; q k h ~B ; k ; d k ( k ; k; ) i where k = M k ; F q k = kb; k; k M k k k B; ; d k; k k k B; d k = q k : k k; k B;

21 MULTILEVEL PRECONDITIONERS FOR MIXED METHODS Theorem. The number of operations for solving system (.) by method (.9) with the matrix ~ B dened in(.7) and with accuracy in the sense (:) k K+ ; k M k ; k M < ; is estimated byqn = ln, where = M ; F, IR N is any initial vector, and the constant Q > does not depend on N. Proof. It is known [] that for a given >, to achieve the accuracy (.) the number of iterations K can be estimated by the inequality K ln (=) ln q where q = ( p ; )=( p +) and is dened in (.8). Thus it is easy to see that the arithmetical cost of the procedure (.9) for solving (.) is approximately (ln (=)=ln q ) times the cost per iteration. The cost per iteration is O(N = )bythe method of separation of variables (.) and (.8), so the desired result follows from Theorem 9.. Preconditioners for a general case. In this section we consider the case where the coecient a is a full tensor and the domain satises the assumption that there is an orientation-preserving smooth mapping L from the unit cube ^ onto and there are positive constants r and Q such that (:) r ; kj (x)k Q 8x ^ and (:) rkj ; (x)k Q 8x where J (x) isthe Jacobian matrix of L at x and kk denotes the matrix -norm. Note that the domain is of size r. Next, we consider the denition of the nonconforming nite element space. Let C^h, P^h,andT^h be the partitions of ^ into cubes, prisms, and tetrahedra, respectively, associated with the mesh size ^h, as dened in x., and let N^h be the P nonconforming space associated with T^h, as given in (.). Set h = r^h and dene N h = ' = L ; : N^h : Also, we introduce the mapping I : N h!n^h dened by Iv = v L. We now dene the stiness matrix M on the domain by (:) (Muv) N = a h (u h v h ) 8u h v h N h where (:) P a h (u h v h ) = T T h ( ; h ru h rv h ) T P = T T^h(jdet(J )j ; h J ; riu h J ; riv h ) T where jdet(j )j is the Jacobian of the mapping.

22 Z. CHEN ET AL. For each cube C C^h,weintroduce the diagonal matrix ; C =diagfa Ca C a C g with some as yet unspecied constants a ic, i =. Then we dene X! X (:) b h (u h v h )= ( ; C ru h rv h ) T 8u h v h N^h CC h C T C where the constants C are scaling factors. One reasonable choice is to take C = ( C + C )=, where C and C are the largest and smallest eigenvalues of the eigenvalue problem (:) ~ ; h (x )v = C C v v IR ; where ~ ; h = jdet(j )j J ; T ; h J ; and x L(C) is some point. Note that the assumptions (.) and (.) imply that there are two constants Q and Q independentofr and h such that (:7) Q a h (u h u h ) rb h (Iu h Iu h ) Q a h (u h u h ) 8u h N h : We consider two useful choices of the matrix ; C :. ; C = I, 8C C h, i.e., the matrix ; C is the identity matrix.. ; C =diagf~; h (x )g, 8C C h, i.e., the matrix ; C is the diagonal part of ~ ; h (x ) at some point x L(C). Note that in both cases the ; constants Q and Q in (.7) only depend on the local variation of the coecients ~ ; h. Hence the problem of dening a preconditioner kl for a h ( ) is reduced to the problem of nding a preconditioner for rb h ( ), which has a diagonal coecient tensorandis dened on the unit cube ^. Namely, all the analyses in x can be carried out here.. Results of the numerical experiments. In this section the method (.9) is tested on the model problem P ; i= @x i a i = f in We presenttwo numerical examples. In the rst example, the domain is the unit cube: =(). The domain is divided into n cubes (n in each direction) and each cube is partitioned into tetrahedra. The dimension of the original algebraic system is N = n ; n. The right hand side is generated randomly, and the accuracy parameter is taken as = ;. The condition number of the matrix B ; M is calculated by the relation between the conjugate gradient and Lanczos algorithms []. The coecients a i, i =, are constants on each cube. The results are summarized in Table, where Iter and Cond denote the iteration number and condition number, respectively. FromnTable we see that the condition number depends on the maximal ratio =max a a CC a a o. The case of < has a better convergence than the case of the h Poisson equation (i.e., a = a = a =). In the second example we treat the Poisson equation on the domain as shown in Figure. The domain is subdivided into 9 9 cubes and the number of unknowns is then N = 9. This problem is solved with accuracy = ;. Twenty iterations are needed to achieve the desired accuracy, and the computed condition number of the matrix B ; M is equal to ten.

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