S MALASSOV The theory developed in this paper provides an approach which is applicable to second order elliptic boundary value problems with large ani

Transcription

1 SUBSTRUCTURNG DOMAN DECOMPOSTON METHOD FOR NONCONFORMNG FNTE ELEMENT APPROXMATONS OF ELLPTC PROBLEMS WTH ANSOTROPY SY MALASSOV y Abstract An optimal iterative method for solving systems of linear algebraic equations arising from nonconforming nite element discretizations of second order elliptic boundary value problems with anisotropic coecients is constructed and studied The technique suggested is based on decomposition of the original domain into nonoverlapping subdomains t is assumed that the equation coecients in each subdomain vary inconsiderably This approach combines the ideas of domain decomposition methods, the algorithms of multilevel and algebraic multigrid methods with the bordering method An iterative process whose convergence rate is independent of the mesh step size and the ratio of anisotropy in the coecients is constructed t is shown that the number of arithmetic operations required for realization of the method with a given accuracy is proportional to the number of unknowns of the original algebraic problem Key words Second order elliptic problem, nonconforming nite element method, domain decomposition method, algebraic substructuring preconditioner, separable matrix, condition number AMS subject classications 5N, 5N, 5F n this paper we proposeaniterative method for solving systems of grid equations approximating boundary value problems for second order elliptic equations with anisotropic coecients The technique suggested is based on decomposition of the original domain composed of rectangles into nonoverlapping subdomains inside which the equation coecients vary inconsiderably At present, there are numerous results on non-overlapping domain decomposition (see, eg, [,,,, 8]) Almost all of them can be viewed as so-called additive Schwarz method [5] Algorithms of this kind, which have satisfactory convergence properties for the case of many regions, have one feature in common n addition to subspaces and subproblems directly related to individual subdomains, there is a global, coarse subspace The absence of such a subspace in general results in slow convergence [5, ] All such methods are based on decomposing the domain into subdomains of size d and involve the solution of related problems on the subdomains and lower-order coupling systems on the subdomain boundaries The best condition number for the preconditioned system is shown to be on the order of ( + ln (d=h)), =, ([8, 5, 5]), where h is the mesh-size parameter Another approaches consist in developing the bordering method [, ] or capacitance matrix method [5,, ] The main idea is to use well known techniques to solve or precondition the problems in the subdomains For the problem at the interfaces a preconditioner is constructed in the form of an inner Chebyshev iterative procedure More precisely, the preconditioner is constructed for the Schur complement of the original symmetric positive denite matrix, dened after eliminating the blocks corresponding to the unknowns in the subdomains [9, ] This approach for solving systems of mesh equations combines the ideas of domain decomposition methods [, 9,,, 8], the algorithms of multilevel and algebraic multigrid methods [,, 9], with the bordering method [, ] The work in this direction has been done in [9, 5,,,,, ] Preconditioning the Schur complement on the interface between subdomains makes it able to construct such methods that the condition numbers for the resulting preconditioned systems are independent of mesh-size parameter h and size of the subdomains d Although the results obtained in all these references include the cases of large jumps in the coecients of the problem, none of them deals with the case of anisotropic coecient tensor y This work was supported in part by the US Department of Energy under Grant #DE-FG5-9ER5 nstitute for Scientic Computation, Texas A&M University, 55 Blocker Bldg, College Station, TX 8{ (malyasov@isctamuedu)

2 S MALASSOV The theory developed in this paper provides an approach which is applicable to second order elliptic boundary value problems with large anisotropy ratio in the coecients The main idea we follow here is close to the construction considered in [9, ] n these works along with the Schur complement on the interfaces S authors introduced some auxiliary interface operator B ~ such that the computation of its inverse is very cheap and the condition number of the preconditioned operator Cond( B ~ S ) is bounded from above by C h, with a positive constant C and h is a mesh-size parameter Then the preconditioner B for operator S is dened in the form of matrix polynomial of degree m: B = P m( ~ B S )S : Using Chebyshev polynomials it is shown that the preconditioner B can be constructed to be spectrally equivalent to the Schur complement S and condition number for the resulting preconditioned system is independent of the diameters of the subdomains d and mesh-size parameter h with only a slight increase in computational eort The arithmetical cost of solving the problem with matrix B is proportional to the dimension of the initial discrete problem, ie O(h ) The purpose of this paper is to develop a variant of the bordering method for second order elliptic partial dierential equations with anisotropy inthe coecients approximated by the nonconforming P elements We note that an important dierence between nonconforming an conforming cases is that there are no nodes at the vertices (or wire basket) of the subregions Thus, there is no problem of crosspoints which isanessential part of the domain decomposition method when using conforming nite elements [8, 5,,,] t makes the construction of the preconditioner for the Schur complement on the interfaces between subdomains very clear and easy The construction of this interface preconditioner is the most interesting part of the work The main result of this paper is that the condition number of the preconditioned operator is bounded by a constant independent of the mesh-size parameter and the coecients of the problem The outline of this paperisasfollows n Section we pose the problem, give itsnonconforming nite element discretization, the matrix formulation and outline the construction of a block diagonal preconditioner to the algebraic system n Section we consider a model problem in a unit square with constant coecients and construct the preconditioners for the problems in subdomains Section contains the main result of the paper t is subdivided into three subsections n the rst subsection we construct a preconditioner for the problem at the interface in the form of an inner iterative procedure considering the union of two rectangular subdomains The second subsection describes an algorithm for implementing the interface preconditioner Then, in the third subsection we construct the interface preconditioner for a general domain composed of rectangles The arithmetic cost of solving the system with the preconditioner proposed is proportional to the number of unknowns of the original algebraic system, ie the preconditioner constructed is of the optimal order of arithmetical complexity n Section we provide the results of numerical experiments These computations show that the theoretical estimates are fully realized in practice Problem Formulation Let be a bounded domain on a plane R,which is composed of open rectangles i whose sides are parallel to the coordinate axes, = m i S Consider an elliptic problem i= () r ( K ~ ru)+~c u = f in u = on ( Kru ~ n) = on where ~ K(x) is a positive denite symmetric coecient matrix, ~c (x) is a nonnegative bounded function, f(x) L () is a given function, [ and \ = We consider the case of = The pure Neumann problem ( = ) can be treated in a similar way but for the sake of simplicity is not described here

3 DOMAN DECOMPOSTON METHOD FOR ANSOTROPC PROBLEMS Assume that the interior of each side of rectangles i either entirely belongs to or, or lies inside Also assume that i, i = ::: m,canhave either a common side or only a common vertex, or they do not overlap t is obvious that any domain composed of rectangles can be partitioned by additional lines into subdomains i satisfying this assumption Let the bilinear form ~a( ) be dened by ~a(u v) =(~ Kru rv)+(~c u v) u v V () = fv H () : v = on g where ( ) denotes the inner product in L () Assume that there a diagonal coecient matrix K(x) = diagfk x (x) k y (x)g and a piecewise constant function c (x), such that k x (x) =k x i k y (x) =k y i c (x) =c i x i i == ::: m with constants k x i >, k y i >, c i, satisfying inequalities () ((Kru ru)+(c u u)) a(u u) ((Kru ru)+(c u u)) for any u V () with some positive constants The standard weak form of () is: nd u V () such that () ~a(u v) =(f v) 8v V (): Let C h be a rectangular mesh in Assume that in each rectangle i the mesh steps h x i, h y i, i = ::: m, are constant in each direction, and the i of the rectangles belong to the mesh lines Also assume that there exist constants c and c independent ofh such that c h min fh x i h y i g max fh x i h y i gc h: i= ::: m i= ::: m Here h == p M, where M is the number of mesh nodes belonging to n Let T h be a regular partitioning of C h into triangles [] and let V h () be the P { nonconforming nite element space of functions v L () []: that is vj are linear for all T h, v are continuous at the middle points of the sides of T h, and vanish at the middle points of the sides of triangles on Note that the space V h () is not a subspace of H () Dene the bilinear forms on V h () by () ~a h (u v) =P T h ( ~ Kru rv) +(~c u v) 8 u v V h () a h (u v) =P T h (Kru rv) +(c u v) 8 u v V h () where ( ) is the inner product in L (), T h Then the P {nonconforming nite element discretization of () is: nd u h V h such that (5) ~a h (u h v)=(f v) 8v V h (): Once a basis f' i (x)g N i= for V h() is chosen, where N = dim V h (), then (5) leads to a system of linear algebraic equations Write u(x) = P N i= u i' i (x) Then () becomes NX i= or in matrix representation u i ~a h (' i ' j )=(f ' j ) j = ::: N () ~Au = f where ~ Aji =~a h (' i ' j ), f j =(f ' j ), i j = ::: N

4 S MALASSOV n the same way we dene matrix A by A ji = a h (' i ' j ), i j = ::: N Then () implies that () (Au u) ( ~ Au u) (Au u) 8u R N ie matrices ~ A and A are spectrally equivalent The underlying method to solve () is a preconditioned iterative method nequalities () suggest considering matrix A as a preconditioner to ~ A Therefore, we need to nd an ecient method for solving the problem (8) Av = g: Let u (i) and v (i) denote the vectors corresponding to the nite element functions u and v from V h ( i ), respectively Let A (i) denote the local stiness matrix arising from a h i ( ): (9) ( ~ A (i) u (i) v (i) )=~a h i (u v) 8u v V h ( i ): For each subdomain i, i = ::: m, we can partition the degrees of freedom u (i) into two sets The rst set includes the degrees of freedom at the nodes in the interior of subdomain i, denoted u (i), and the second set corresponds to the degrees of freedom at the nodes on the i n, denoted u (i) Such a partitioning induces the partitioning of A(i) given by " #" # " #! (A (i) u (i) v (i) A (i) )= A (i) u (i) v (i) () : A (i) A (i) u (i) v (i) () Finite element system (8) has the obvious algebraic representation: A () with block A dened by () A () A (m) A (m) A () A (m) A (A u v )= 5 mx v () v (m) v 5 = (A (i) u (i) v(i) i= ): g () g (m) g Note that blocks A (i), i = ::: m, correspond to the boundary value problems in the rectangles i 5 () a h i (u h v)=g(v) 8v V h ( i ) i = ::: m with homogeneous Dirichlet boundary conditions imposed on the i Denote the m number of degrees of freedom in i n, [ i and m i n by N (i) i= i=, N (i), N and N, i = ::: m, respectively Eliminating the unknowns v (i), i = ::: m, in (), we obtain the following Schur system: () where (5) = A mx i= h A (i) A (i) v = G i A (i) G = g mx i= h A (i) A (i) i g (i) :

5 DOMAN DECOMPOSTON METHOD FOR ANSOTROPC PROBLEMS 5 Thus, The solution to system () can be reduced to the construction of an ecient algorithm for solving systems () in subdomains and the Schur complement system () The algorithm for solving subdomain problems with matrices A (i) is considered in Section t is shown that these problems can be solved very eciently The main goal of this paper is to construct an easily invertible matrix B spectrally equivalent to matrix : c (B v v ) ( v v ) c (B v v ) 8v R N where constants c and c are independent of the mesh size parameter h, the subdomain diameters, and value of the coecients This issue is discussed in detail in Section Model problem in a unit square Here we discuss an algorithm to solve the subdomain systems Consider a model problem on a unit square: u k u + c u = f u = in [ ] where coecients k x >, k y >, c, are constants in t is clear that a method developed for this model problem can be easily generalized for the case of rectangular domain and mixed boundary conditions Let C h = fc (i j) g be a partition of into uniform squares with the length of the side h ==n, where (x i y j ) is the lower left corner of the square C (i j) We enumerate the squares in a lexicographical order, rst, in the y-direction, then in the x-direction Next, we divide each square C (i j) into triangles as shown in Figure a The partitioning of into triangles is denoted by T h We introduce the set of centers of all edges of the triangulation of, and the set Q h of those centers that are not on the Dirichlet boundary (see Fig a) The Crouzeix- Raviart P {nonconforming nite element space V h is dened by () V h = fv L () : Let the dimension of V h be N Obviously, N n vj P () 8 T h v is continuous at the points from Q h and vanishes at the middle points of edges on g: 5 (a) Triangilation of the domain (b) Local enumeration of the degrees of freedom () Figure Triangilation and partition of the degrees of freedom Now we dene the bilinear form on V h by a h (u v) = X T @y + c uv dx 8 u v V h : Thus the nonconforming discretization of problem () is given by seeking u h V h such that () a h (u h v)=(f v) 8 v V h :

6 S MALASSOV For any function v V h we denote by v R N its representation with respect to the basis in V h Let (u v) N be a standard bilinear form dened on R N by(u v) N = P xq h u(x)v(x), 8u v V h Then dene symmetric and positive denite operator A :R N! R N by (5) (Au v) N = a h (u v) u v V h : For each square C = C (i j) C h, denote by Vh C the subspace of the restriction of the functions from V h into C For each v Vh C, we indicate by v c the corresponding vector The dimension of Vh C is denoted by N c Obviously, for a square without faces on wehave N c =5 The local stiness matrix A C on a square C C h is given by () (A C u c v c ) Nc = P C k ) + k ) + c (u v) 8u v V C h : Note that matrices A C are positive denite when C \ = and at least semidenite otherwise (if c = then all the matrices A C are positive denite) The global stiness matrix is determined by assembling the local stiness matrices: X () (Au v) N = (A C u c v c ) Nc 8u v R N : CC h To dene the solution procedure we divide all the unknowns in the system into two groups: The rst group consists of the unknowns corresponding to the edges of the triangles that are internal for each square (these are unknowns corresponding to the nodes marked by \" in Fig ) We denote these unknowns by vc (i j), i j = ::: n The second group consists of all unknowns corresponding to the edges of the squares in the partition C h, without the faces on (Fig, the nodes marked by \") (a) First, we enumerate the unknowns on the edges perpendicular to the x-axis (nodes and in Fig b) We denote these unknowns by vx (i j), i = ::: n, j = ::: n (b) Second, we enumerate the unknowns on the edges perpendicular to the y-axis (nodes and 5 in Fig b) We denote these unknowns by vy (i j), i = ::: n, j = ::: n Now we consider a square C that has no face on the and enumerate the edges s j, j = ::: 5, of the triangles in this square in correspondence with the partitioning introduced above as is shown in Figure b Then the local stiness matrix of this square has the following form: A C A c A (8) = c : A c A c ntroducing parameter c = c h = we can write (9) A c =[k x + k y + c] A c = A T c =[k x k x k y k y ] A c = k x k x k y k y 5 +c 5 :

7 DOMAN DECOMPOSTON METHOD FOR ANSOTROPC PROBLEMS The splitting of space R N induces the presentation of the vectors: v T =(v T vt ), where v R N, v R N, and v corresponds to the unknowns of the -nd group Obviously, N = n and N = N n Then matrix A can be presented in the following block form: () A = A A A A where A :R N! R N is a diagonal matrix Now denote by ^A = A A A A the Schur complement ofa obtained by elimination of the vector v Then A = ^A + A A A, so matrix A has the form () A = ^A + A A A A : A A To understand the structure of the Schur complement ^A let us write explicitly the matrix equation Av = g in terms of the unknowns vc i j, vx i j,andvy i j For any square C (i j) = wehave: (k x + k y + c)vc i j k x (vx i j + vx i+ j ) k y (vy i j + vy i j+ )=gc i j i j = ::: n (k x + c)vx i j k x (vc i j + vc i j )=gx i j i = ::: n j = ::: n (k y + c)vy i j k y (vc i j + vc i j )=gy i j i = ::: n j = ::: n: After eliminating the unknowns vx i j and vy i j wehave a 5-point computational scheme for the unknowns vc i j : () where () (a x +a y + b)vc i j a x (vc i j + vc i+ j ) a y (vc i j + vc i j+ )= ~gc i j a x = k x +c=k x a y = k y b =c + +c=k y +c=k x + : +c=k y t is easy to see matrix ^A can be represented in a tensor product form (according to the enumeration introduced earlier in this section): () ^A = a x (A x y )+a y ( x A y )+b( x y ) where the matrices x y :R n! R n are identity ones, and the matrices A x A y :R n! R n are tridiagonal: (5) A x = A y = To solve the problem with separable matrix ^A we can use either the discrete fast Fourier transform [] or an algebraic multigrid method (AMG) [,, 9,, 8] When an implementation cost of the rst method is estimated by O(h ln (h )), the AMG methods have the optimal order of arithmetic complexity O(h ) Since these methods are well described in the literature we are not going to discuss them in greater detail 5 :

8 8 S MALASSOV Preconditioner for the interface problem n this section we construct a preconditioner for the problem at the interface in the form of an inner iterative procedure More precisely, we construct here a preconditioner for the Schur complement of the original matrix Model problem For the sake of simplicity, let us consider the problem () for the case of being a rectangle composed of two squares = [,= \ (see Fig ): () =f(x y) :<x< <y<g i = f(x y) :i <x<i <y<g i = : Assume that coecients k x, k y, c are constants in i, i =, ie k x k x i = const > k y k y i = const > c c i = const : Let T h be a regular triangulation of domain with mesh-size h (as described in Section ) n this case () has the form: () A () A () A () A () A () A A() 5 v() v () v 5 = g() g () g Note that blocks A (i), i =, correspond to the boundary value problems in subdomains i and block A is a diagonal matrix 5 : (a) Triangilation of the domain (b) The degrees of freedom of the reduced problem Figure Degrees of freedom of real and reduced problems Eliminating the unknowns vx i j and vy i j in each subdomain as is discussed in Section we get the problem () ^A () ^A() ^A() ^A () ^A () ^A () A 5 v() c v c () v 5 = g() c g c () g where blocks ^A(i), i =, are separable matrices, and vectors v(i) c, i =, consist of the unknowns vc i j in each subdomain n Figure b, the nodes corresponding to these unknowns are marked by \" Matrix A is dened by equality () (A u v )= ntroducing the subdomain Schur complements X i= (A (i) u(i) v(i) ): 5 () (i) = A(i) ^A(i) h ^A(i) i ^A(i) i =

9 DOMAN DECOMPOSTON METHOD FOR ANSOTROPC PROBLEMS 9 the Schur complement (5) can be rewritten in the form: (5) () = A X i= ^A (i) h i ^A(i) ^A(i) =() +() : We construct a preconditioner B for by dening preconditioners B () and B () for and (), respectively, and setting () B = B () + B() : Let us consider subdomain, matrices () and A (), and omit the index \()" for simplicity Boundary nodes belonging to (marked by \") and internal nodes (marked by \") are schematically shown in Figure The following lemma is valid Lemma There existsanh-independent constant such that () h (A u u ) ( u u ) (A u u ) 8u R N : Proof Consider an eigenvalue problem (8) u = A u u R N : Since the symmetric matrices and A are positive denite problem (8) has N positive eigenvalues Figure Degrees of freedom of the model subdomain problem t is obvious that eigenvalues i, i = ::: N, of problem (8) can be found from the system of equations (9) ^A u + ^A u = ^A u + A u = A u : Here matrix ^A is dened by () The N N matrix ^A has a form ^A = k x y 5 k x 5 y and the diagonal N N matrix A is dened by A =(k x + c) y Dene by l R n an eigenvector of nn matrix A y (5) corresponding to the eigenvalue l : () A y l = l l l = ::: n:

10 S MALASSOV Note that we explicitly know the eigenvalues and eigenvectors of matrix A y : n () sin l = ::: n: l = sin l n l = l(i ) n i= Fix some l [ n] and dene a vector u v l = u l with some vector v R n and substitute it in system (9) From the second equation of (9) it follows that eigenvalues of problem (8) are dened by the expressions () l = k x k x + c v(l) n l = ::: n where a parameter v (l) n or, using expressions (), is the n-th component ofvector v (l) from the system ~A (v (l) l )= ~ A l () (a x A x +( l a y + b) x ) v (l) =k x ::: T : () ntroducing notations: d l = l a y a x + b a x e = k x + c a x k x system () can be rewritten in the form: (5) +d l +d l +d l +d l 5 v (l) v (l) v (l) n v (l) n 5 = e 5 : Set x l =+ d l and consider the following recurrent sequence: () = = x l + i+ = x l i i i = ::: n : t is easy to see that i = U i (x l )+U i (x l ) i = ::: n where U m (x) is the Chebyshev polynomial of the -nd kind of degree m: U m (x) = p x By induction it can be shown that x + p x m+ x + p x (m+) : () v (l) n = e n U n (x l )+U n (x l ) = e n + n U n (x l )+U n (x l )+U n (x l ) :

11 DOMAN DECOMPOSTON METHOD FOR ANSOTROPC PROBLEMS Since x l then v n (l) for any l = ::: n From () it follows that the maximal eigenvalue of problem (8) is bounded from above by max Nowwehave to estimate the minimal eigenvalue min of problem (8) from below Taking into account that U n (x l )=x l U n (x l ) U n (x l )we get (8) Since v n (l) = e (x l +)U n (x l ) U n (x l ) = e (x l +)U n (x l ) xl + x l + U n (x l ) (x l +)U n (x l ) : U n (x) U n (x) = (x + p x ) n+ (x + p x ) (n+) (x + p x ) n (x + p = p x ) n = x + x + (x + p x ) n from (), (), and (8) it follows that (9) l = = r xl x l + r dl +d l p n x l + x l + d l + q d l + d l C A = To estimate expression q (9) from below we consider two cases: y d l + d l + d l =n Then we have n(n+) n C A : t means that d l + p d l ( + d l ) =n, or d l r dl l p = +d l +8n(n +) n + h 5 : y<=n n this case d l < So, (9) is estimated from below as follows n(n+) l = = r dl +d l + r dl + +d n + +d l r dl min n ( + y) n (e )ny! d l + p d l ( + d l ) d l + p d l ( + d l ) h : r dl +d l + r dl +ny = +d l ny A = n + + e ny p d l + d l ( + d l ) d l + p d l ( + d l ) From these estimates it follows that l h=5 forany l = ::: n Thus, the minimal eigenvalue of problem (8) is bounded from below by min h=5 andwehave h 5 (A() u u ) ( () u u ) (A () u u ) 8u R N :

12 S MALASSOV Analogously, wehave the same estimates for matrices A () and (), which completes the proof Remark Lemma holds true if on some of the other edges of the rectangular subdomain a homogeneous Neumann boundary condition is imposed Remark f a homogeneous Neumann boundary condition is imposed on the whole remaining part of the boundary of n then inequalities () of Lemma should be replaced by: () h (A u u ) ( u u ) (A u u ) 8u R N n ker ( ) where constant does not depend on mesh size parameter h and coecients of the subdomain problems We proceed with the construction of preconditioner B We dene it in the form of an inner Chebyshev iterative procedure [, 9, 9] From Lemma we know that the eigenvalues of matrix A belong to segment [h=5 ] Let P L (y) be the polynomial of least deviation from zero on this segment that satises the condition P L () = Denote by l, l = ::: L, the inverses of the roots of polynomial P L (y) The formulae for P L (y) and its roots = l, l = ::: L, can be found, eg in [] Then preconditioner B for matrix is determined by: () B = ( LY l= ) l A : The procedure for calculating the vector w = B g for given g R N has the form: () w () = w (l) = w (l) l A w = w (L) : w (l) g l = ::: L For computational stability, instead of (), we can use the three-term recurrence relation [] We return to the realization of the iterative procedure () in the next subsection Lemma and the theory of Chebyshev iterative methods imply the following basic result Theorem Let L (5=h) = Then matrix B in () is spectrally equivalent to matrix with constants of equivalence independent of mesh size parameter h and the value of coecients k x i, k y i, c i, i =, in the subdomains Remark Clearly, in the theory L is chosen to be of the order (5=h) = n practice it is calculated explicitly after the boundaries of the spectrum of matrix A have been calculated by an appropriate iterative procedure [] Arithmetical complexity of the interface preconditioner n order to estimate the arithmetic complexity ofmultiplying avector by matrix B it is sucient to estimate the arithmetic complexity ofmultiplying a vector by matrix because we know that matrix A is diagonal and the number of iterations L in the inner Chebyshev iterative procedure () is of the order (5n) = The procedure of nding the product u G is based on a partial solution technique [,,8], whichwe outline here Let us consider the terms () u and () u in the expression v = () u + () u separately Below we dene the multiplication procedure only for the rst term () u The second term is treated in the same manner Again, we omit the index \()" for simplicity

13 DOMAN DECOMPOSTON METHOD FOR ANSOTROPC PROBLEMS First, vector v = u = A u ~ A ~ A ~ A u is rewritten as () v = A u + ~ A u where vector u is dened from the system of equations () ~A u = ~ A u k x u with matrix ~ A dened by () Let us denote by W y an orthogonal n n matrix of eigenvectors of problem () and by L y a diagonal n n matrix of the eigenvalues of matrix A y (): 5 W y = f ::: n g L y = diag f ::: n g : Then we dene an N N orthogonal matrix Q = x W y ntroducing a vector v = Q T u and multiplying both parts of equation () by matrix Q T we get the following matrix equation: (5) Q T ~ A Qv (a x (A x y )+a y ( x L y )+b( x y )) v =k x Note that this system can be decoupled into n independent linear systems: () (a x A x +(b + l a y ) x ) v (l) v (l) n v n (l) 5 = k x w l 5 l = ::: n v (n) n 5 : W T y u where the component w l is the l-th component of the backward Fourier transform w = Wy T u of vector u As soon as we ndvector v from systems () we compute the product v ṇ () () ~A u = A ~ Qv = k x W y From () it follows that to dene product u we need to know only the last components v n (l), l = ::: n, of the solution vectors v (l) from systems () Now we dene the partial solution algorithm: On the initial step we solve n systems with three-diagonal n n matrices: (a x A x +(b + l a y ) x ) x (l) x (l) n x (l) n and store the last components of the vectors x (l),ieparameters x (l) n, l = ::: n t requires O(n ) operations and O(n) elements of memory 5 = 5 5 :

14 S MALASSOV Given vector u we use the discrete fast Fourier transform algorithm to compute a vector w =k x Wy T u for only O(n ln n) operations Then, we compute a vector v = h v () n ::: v (n) n i from () by the formulae v (l) n = w l x (l) n, l = ::: n Again, we use the discrete fast Fourier transform algorithm to compute a vector p = k x W y v Finally, we compute the vector v = A u + p As a result of this algorithm the procedure of multiplying a vector by matrix B can be implemented for O(n + n = ln n) operations Note that this estimate does not depend on the coecients of the problem Remark We provided computations of the complexity for so-called \parallel" partial solutions, that is, when the grid line where we need to nd a solution is parallel to the grid line where the right-hand side is nonzero For the case of so-called \perpendicular" partial solutions, when the grid line where we need the solution is perpendicular to the grid line with the nonzero right-hand side we can use an algorithm of the approximate partial solution [, 8,] nstead of computing the exact value of v = u,we calculate the approximation v (") =(") u t can be shown [, ] that taking the accuracy parameter " h p, p, the matrix (") is spectrally equivalent to As a consequence one can develop the partial solution algorithm for the general case when we need to nd the partial solution on the entire boundary of the rectangular subdomain Using the results of [, 8] the partial solution on the i of the rectangular subdomain i can be found for O(n +pn ln (n)), where the rst term of this estimate corresponds to the initial step and can be obtained before starting the iterative process f we use the algorithm of the approximate partial solution developed in [] we have an estimate of computational cost O(n + pn = ln (n)) Therefore, the algorithm of multiplying a vector by matrixb can be implemented for O(n + pn = ln (n)) operations This estimate does not depend on the coecients of the problem nterface preconditioner for general problem Now let us consider problem S () in the domain = m i, being the union of m rectangles i, i = ::: m, as is i= described in Section Using the same arguments as in Subsection it is easy to show that the statement of Lemma holds true even for a general domain composed of rectangles Since preconditioner B is constructed by dening subdomain preconditioners B (i) it is sucient toconsider the model problem in the rectangular subdomain i with homogeneous Neumann boundary S condition on the whole = i Boundary nodes belonging to (\") and i= internal nodes (\") are schematically shown in Figure Figure Degrees of freedom of the Neumann subdomain problem Denote by u and by u i, i = :::, the vectors of the degrees of freedom on the

15 DOMAN DECOMPOSTON METHOD FOR ANSOTROPC PROBLEMS 5 boundaries and i, i = :::, respectively Following [], one can show that for any vector v R N such that v? ker ( ) the following is valid: (8) ( v v )= inf v h V h () v h j =v h a h (v h v h ) inf a w i hv h () wi hj i =vh i h (w h i w h i )=( i v i v i ) i = ::: where u h, uh i, i = :::, are piecewise constant functions dened on, i, i = :::, and generated by vectors u, u i, i = :::, respectively From (), (), and (8) it follows that for any vector v R N such that v? ker ( )wehave (9) (A u u ) ( u u ) h P i= P i= ( i u i u i ) ( ^Ai i u i u i ) h (A u u ): t means that we can dene preconditioner B for the Schur complement (5) in the form () By Theorem matrix B is spectrally equivalent to matrix provided that the degree L of the matrix polynomial () is chosen to be O(h = ) Using the partial solution technique described in Subsection and Remark one can shown that the procedure of multiplying a vector by matrix B can be implemented for O(h + h = ln h ) Now assume that we use the AMG methods to solve the subdomain problems Then problem (8) with matrix ^A can be solved for O(h + h = ln h ) operations As was mentioned in Section we use the matrix A as the preconditioner in a preconditioned iterative method to solve the problem () Summarizing the results of Sections and we can formulate the following statement Statement Under the above assumptions the arithmetic complexity of the proposed algorithm for solving problem () is estimated from above by C (h + h = ln h ) where constant C is independent of mesh size parameter h and coecients of the problem ~ K(x) and ~a (x) Results of the numerical experiments n this section the method of preconditioning being presented is tested on the model problem in the unit u k u = f in [ ] u = Domain is composed of subdomains as shown in Figure 5 Coecients k x and k y are constants in each subdomain k x = k y = k k x = k k y = k x = k k y = k x = k y = k Figure 5 Coecients in the subdomains for a model problem

16 S MALASSOV The domain is divided into n squares (n in each direction) and each square is partitioned into triangles The dimension of the original algebraic system is N = n n and the dimension of the Schur complement after elimination of the subdomain problems is N =n The right-hand side is generated randomly, and the accuracy parameter is taken as " = The degree of matrix polynomial () equals L = p :5n +,where[] isaninteger part of The condition numberofmatrixa is calculated by the relation between the conjugate gradient and the Lanczos algorithm [] The results are summarized in Table, where n iter and Cond denote the iteration number and condition number, respectively All experiments are carried out on Sun Workstation Table N = 98 N =9 N = 9 k n iter Cond n iter Cond n iter Cond Acknowledgment The author is grateful to Professors Raytcho Lazarov and Jim Bramble for valuable comments and suggestions and to the seminar of Numerical Analysis at Mathematical Department oftexas A&M University for possibility to present and discuss the results of this paper This research has been part of the Partnership in Computational Sciences (PCS) Project sponsored by the US Department of Energy under Grant #DE-FG5-9ER5 REFERENCES [] D Arnold and F Brezzi, Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates, RARO Model Math Anal Numer, 9 (985), pp { [] O Axelsson and P Vassilevski, Algebraic multilevel preconditioning methods,, Numer Math, 5 (989), pp 5{ [] N Bakhvalov and M Orekhov, Fast methods of solving Poisson's equation, USSR J Comput Maths Math Phys, (98), pp { [] A Banegas, Fast Poisson solvers for problems with sparsity, Math Comp, (98), pp { [5] P Bjrstad and O Widlund, terative methods for the solution of elliptic problems on regions partitioned into substructures, SAM J Numer Anal, (98), pp 9{ [] J Bourgat, R Glowinski, P Tallec, and M Vidrascu, Variational formulations and algorithms for trace operator in domain decomposition calculations, in Second nt Symp on Domain Decomposition Methods for PDEs, T Chan, R Glowinski, J Periaux, and O Widlund, eds, Philadelphia, PA, 989, SAM, pp { [] J Bramble, J Pasciak, and A Schatz, The construction of preconditioners for elliptic problems by substructuring,, Math Comp, (98), pp { [8], The construction of preconditioners for elliptic problems by substructuring,, Math Comp, 9 (98), pp { [9], The construction of preconditioners for elliptic problems by substructuring,, Math Comp, 5 (988), pp 5{ [] J Bramble, J Pasciak, and J Xu, The analysis of multigrid algorithms with non-nested spaces or non-inherited quadratic forms, Math Comp, 5 (99), pp { [] P Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 98 [] M Dryja, A nite element capacitance method for the elliptic problem, SAM J Numer Anal, (98), pp {8 [], A nite element capacitance method for elliptic problems on regions partitioned into subregions, Numer Math, (98), pp 5{8

17 DOMAN DECOMPOSTON METHOD FOR ANSOTROPC PROBLEMS [] M Dryja and O Widlund, Some domain decomposition algorithms for elliptic problems, in terative Methods for Large Linear Systems, L Hayes and D Kincaid, eds, Academic Press, San Diego, CA, 989, pp {9 [5], Towards a unied theory of domain decomposition algorithms for elliptic problems, in Third nt Symp on Domain Decomposition Methods for PDEs, T Chan, R Glowinski, J Periaux, and O Widlund, eds, Philadelphia, PA, 99, SAM, pp { [] G Golub and C V Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, PA, 989 [] L Hageman and D Young, Applied terative Methods, Acad Press, New York, 98 [8] Y Kuznetsov, Block-relaxation methods in subspaces, their optimization and application, in Numerical Methods in Applied Mathematics, (Paris, 98), G Marchuk and J-L Lions, eds, Novosibirsk, Russia, 98, "Nauka" Sibirsk Otdel, pp 9{ (n Russian) [9], Multigrid domain decomposition methods for elliptic problems, Comput Meth Appl Mech and Eng, 5 (989), pp 85{9 [] S Margenov and P Vassilevski, Algebraic multilevel preconditioning of anisotropic elliptic problems, SAM J Scientic Computing, 5 (99), pp { [] A Matsokin and S Nepomnyaschikh, On using the bordering method for solving systems of mesh equations, Soviet J Numer Anal and Math Modeling, (989), pp 8{9 [] S Nepomnyaschikh, On the application of the bordering method to the mixed boundary value problem for elliptic equations and on mesh norms in W = (S), Soviet J Numer Anal and Math Modeling, (989), pp 9{5 [] T Rossi, Fictitious Domain Methods with Separable Preconditioners, PhD thesis, Department of Mathematics, University of Jyvaskyla, Finland, December 995 [] A Samarski and E Nikolaev, Methodes de Resolution des Equations de Mailles, Mir,Moscow, Russia, 98 [5] M Sarkis, Schwarz Preconditioners for Elliptic Problems with Discontinuous Coecients Using Conforming and Non-conforming Elements, PhD thesis, Courant nstitute of Mathematical Sciences, New York University, New York, September 99 [] P L Tallec, Domain Decomposition Methods in Computational Mechanics, Computational Mechanics Advances, vol, North-Holland, Amsterdam, 99 [] R Varga, Matrix terative Analysis, Prentice Hall, Englewood Clis, NJ, 9 [8] J Xu, terative methods by space decomposition and subspace correction, SAM Rev, (99), pp 58{