MULTILEVEL PRECONDITIONERS FOR NONSELFADJOINT OR INDEFINITE ORTHOGONAL SPLINE COLLOCATION PROBLEMS

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1 MULTILEVEL PRECONDITIONERS FOR NONSELFADJOINT OR INDEFINITE ORTHOGONAL SPLINE COLLOCATION PROBLEMS RAKHIM AITBAYEV Abstract. Efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the the solution of an orthogonal spline collocation (OSC) discretization of a Dirichlet boundary value problem with a nonselfadjoint or an indefinite operator. The OSC solution is sought in the Hermite space of piecewise bicubic polynomials. It is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal OSC equation. The preconditioners are used with the preconditioned conjugate gradient method, and numerical results are presented that demonstrate their efficiency. Key words: Orthogonal spline collocation, multilevel methods, preconditioner, nonselfadjoint or indefinite operator, elliptic boundary value problem. AMS subject classification. 65N35, 65N55, 65F10 1. Introduction. Let Ω be a unit square (0, 1) (0, 1) with boundary Ω, and let x = (x 1, x 2 ). We consider a Dirichlet boundary value problem (BVP) (1.1) where we let Lu = f in Ω, and u = 0 on Ω, (1.2) 2 2 Lu (x) = a ij (x)u xix j (x) + b i (x)u xi (x) + c(x)u(x). i,j=1 i=1 With respect BVP (1.1), we mae the following assumptions. The functions {a ij } 2 i,j=1, {b i } 2 i=1, c, and f are sufficiently smooth, and a 12 = a 21. Differential operator L satisfies the uniform ellipticity condition; that is, there is ν > 0 such that (1.3) 2 a ij (x) η i η j ν(η1 2 + η2), 2 x Ω, (η 1, η 2 ) R 2. i,j=1 For any f L 2 (Ω), BVP (1.1) has a unique solution u(x) in (1.4) H 2 0 (Ω) = {v H 2 (Ω) : v = 0 on Ω}, where L 2 (Ω) and H 2 (Ω) are the Sobolev spaces. We approximate BVP (1.1) by an orthogonal spline collocation (OSC) scheme, in which the discrete solution is sought in the Hermite space of piecewise bicubic polynomials, and it satisfies the differential equation exactly at the special set of collocation points. The primary advantages of the OSC method are as follows: low computational cost of forming a linear system of algebraic equations, relatively easy application of higher order finite elements, the OSC solution possesses optimal order error estimates [7], and the solution exhibits the so-called superconvergence property [8, 15]. The matrix of a linear system resulting from the OSC discretization is sparse and can be expressed as a sum of tensor products of so-called almost bloc diagonal matrices [2]. Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801, U.S.A., aitbayev@nmt.edu

2 2 R. Aitbayev The solution of the OSC equations by a banded Gaussian elimination requires O(N 2 ) arithmetic operations, where N is the number of unnowns [22, 23, 34]. If the differential operator is separable and the partition is uniform in one direction then the OSC problem can be solved by a fast direct algorithm with the cost O(N log N) [10]. Classical iterative methods, such as Jacobi, Gauss-Seidel, and SOR, for the OSC solution of Poisson s equation on a uniform partition were studied in [21, 26, 37]. ADI methods for solving OSC problems with separable operators were investigated in [5, 14, 18]. Modern techniques are underdeveloped for the solution of OSC equations in comparison with finite element Galerin or finite difference methods. Only a few optimal cost algorithms have been proposed to solve the OSC discretization of selfadjoint positive definite BVPs. In [19], the multigrid method was applied to the OSC problem and compared with a multigrid finite difference method. The author concluded that the proposed multigrid OSC method is less efficient than a multigrid finite difference method. A domain decomposition based fast solver for the OSC discretization of the Dirichlet problem for Poisson s equation was developed in [6] which requires O(N log log N) arithmetic operations. In [13], multigrid methods were developed and analyzed for quadratic spline collocation equations. Numerical results were presented indicating that a multigrid iteration is an efficient solver for the quadratic spline collocation equations. It is well nown that the primary issue in the efficient application of an iterative algorithm for solving a BVP is the construction of a good preconditioner. In [24], the authors studied preconditioning of a nonselfadjoint or an indefinite OSC operator by a finite element operator and investigated H 1 condition numbers and the distribution of singular values of the preconditioned matrices. Additive and multiplicative multilevel preconditioners were proposed in [9] for the iterative solution of the OSC discretization of a selfadjoint positive definite Dirichlet BVP. It was proved that the preconditioners are uniformly spectrally equivalent to the OSC operator corresponding to a BVP with the Laplacian and require O(N) arithmetic operations. An efficient two-level domain decomposition type edge preconditioner was proposed in [29] that requires O(N) arithmetic operations. The preconditioner is applied with GMRES method and the number of iterations is independent of the partition stepsize h. Numerical techniques developed for selfadjoint positive definite BVPs are usually inefficient or even fail when applied to nonselfadjoint or indefinite BVPs, and hence, special, more sophisticated, methods are required to obtain the solution [4, 11, 12, 27, 28]. A fast direct preconditioning algorithm for the solution of the normal OSC equation approximating nonselfadjoint or indefinite BVPs was proposed in [1]. In this wor, we develop an additive and a multiplicative multilevel preconditioners for the computation of the solution of the normal OSC equation. Results and algorithms presented in this paper are closely related to those in [1, 7, 9, 31, 32, 39, 40]. To prove uniform spectral equivalence of our preconditioners, we use the approach described in [31] and [32] based on the equivalence of a norm of a certain Besov space with the Sobolev H 2 -norm. We note that the approach used in [39] and [40] requires higher regularity of the solution of BVP (1.1). Our main conclusion from this wor is that the general framewor of multilevel methods can be applied to the OSC discretization of BVPs to construct efficient preconditioners. Rather general nonselfadjoint or indefinite OSC problems can be preconditioned quite well by the proposed multilevel OSC preconditioners.

3 Multilevel preconditioners for OSC problems 3 The outline of this article is as follows. We introduce our notation and define the OSC problem in 2. In 3, we present auxiliary facts that are used to prove main results of this wor. In 4, we define an additive and a multiplicative OSC preconditioners and prove that they are uniformly spectrally equivalent to the operator of the normal OSC equation. In 5, we introduce the matrix-vector form of the OSC problem in the standard Hermite finite element basis and obtain recurrence relations for the computation of the OSC approximations and other quantities required by the multilevel algorithms. In 6, we describe implementations of the additive and the multiplicative preconditioners, and in 7, we present numerical results of the application of the preconditioners with the preconditioned conjugate gradient (PCG) method to solve test problems. 2. OSC problem. In this section, we introduce our notation and define the OSC problem. Throughout this paper, C, C 1, and C 2 C 1 denote generic positive constants independent of the partition stepsize, the number of partion levels, and other variables in the expressions where the constants appear. By L 2 (Ω) and H 2 (Ω), we denote the standard Sobolev norms. Construction of nested spaces. We set π 0 = Ω and, for integer K > 0, we construct a sequence of partitions {π } K =0 by subdividing each rectangular element of partition π 1 into four congruent rectangular elements of partition π. Let h = 2 denote the stepsize of partition π. In what follows, if not stated otherwise, the index variable taes all values in {0, 1,..., K}. We note that there are K + 1 partition levels, and integer K is an important parameter in our analysis. Let V be the vector space of Hermite piecewise bicubic polynomials that vanish on Ω, which has the dimension N = 4 +1 (see Chapter 3 in [35]). The sequence of vector spaces {V } is nested as follows: (2.1) V 0 V 1... V K H 2 0 (Ω). We denote h = h K, N = N K, π h = π K, and V h = V K. Original OSC equation. Let G h be the set of nodes of the two-dimensional composite Gaussian quadrature on partition π h with 4 nodes in each element of π h. In the OSC discretization of BVP (1.1), we see u h V h that satisfies the OSC equations (2.2) Lu h (ξ) = f(ξ), ξ G h, where the differential operator L is defined in (1.2). Existence and uniqueness of a solution and a convergence analysis of problem (2.2) are given in [7]. The OSC problem (2.2) can be written as the operator equation (2.3) L h u h = f h, where the OSC operator L h from V h into V h and the vector f h V h are defined by (2.4) (L h v)(ξ) = Lv(ξ), ξ G h, for any v V h, f h (ξ) = f(ξ), ξ G h. Both L h and f h are well-defined since a function in V h is uniquely determined by its values at G h (see Lemma 5.1 in [33]). We call (2.3) the original OSC equation.

4 4 R. Aitbayev Normal OSC equation. The vector space V h is a Hilbert space with the inner product (2.5) (v, w) h = h2 4 ξ G h v(ξ)w(ξ), which corresponds to the composite Gaussian quadrature on π h. Let L h be the adjoint to L h with respect to the inner product (, ) h. Applying L h on (2.3), we obtain the normal OSC equation (2.6) L hl h u h = L hf h. We introduce a bilinear form (2.7) a h (w, v) = (L hl h w, v) h, w, v V h, and consider the following variational form of problem (2.6): find u h V h that satisfies (2.8) a h (u h, v) = (L hf h, v) h for all v V h. The problems (2.8) and (2.2) are equivalent; hence, problem (2.8) has a unique solution. In this wor, we develop and analyze multilevel preconditioners for the iterative solution of (2.8). Space decompositions. In what follows, we denote and,i for K =0 and K N =0 i=1, respectively, where N is the dimension of V. Let {ψi }N i=1 be a finite element basis of V that satisfies (2.9) C 1 h 1 α α ψ i L 2 (Ω) C 2 h 1 α, α 2, where α = (α 1, α 2 ) is a multi-index (see Theorem 5.7 in [3]). Let (2.10) V i = span(ψ i ), 1 i N, be one-dimensional subspaces of V. Based on (2.1), we consider the following two space decompositions: For v V h, let V 1 (v) = V 2 (v) = { { {v } : {v i } : V = V h and V i = V h.,i } v = v, v V, 0 K, } v i = v, v i V i, 1 i N, 0 K. i We call an element in V 1 (v) and in V 2 (v) a representation of v. The sets V 1 (v) and V 2 (v) will be used to define auxiliary equivalent norms on V h.

5 Multilevel preconditioners for OSC problems 5 3. Auxiliary results. In this section we present auxiliary facts that are used to prove main results of this wor. The following inequalities are proved in Theorem 3.1 in [1]. Lemma 3.1. For h sufficiently small, (3.1) C 1 v 2 H 2 (Ω) a h(v, v) C 2 v 2 H 2 (Ω), v V h. Remar. In what follows, by sufficiently small h, we mean values of h for which the statement of Lemma 3.1 holds. It follows from (3.1) and (2.7) that, for h sufficiently small, the bilinear form a h (, ) is an inner product on V h. The following is the estimate (8.24) in Chapter 3 of [25]. Lemma 3.2. (3.2) v H 2 (Ω) C v L 2 (Ω), v H 2 0 (Ω). Inequalities (3.1), (3.2), and (3.3) v 2 C L 2 (Ω) (vx 2 1x 1 + vx 2 2x 2 )dx, v H 0 2 (Ω), imply that, for h sufficiently small, Ω (3.4) C 1 v 2 L 2 (Ω) a h(v, v) C 2 v 2 L 2 (Ω), v V h. Lemma 3.3. Let v V, and let (3.5) Then, (3.6) N v = c j ψi and c v = (c 1,..., c N ) t. i=1 C 1 h c v v L 2 (Ω) C 2 h c v, v V, where is the 2-norm on R N. Proof. Using (3.5), we obtain v 2 L 2 (Ω) = N c i ψi Ω i=1 N c j ψj dx = j=1 N i,j=1 c i c j Ω ψ i ψ j dx. The inequalities in (3.6) follow from the last identity and the fact that the eigenvalues of the mass matrix corresponding to the finite element basis {ψi }N i=1 belong to the interval [C 1 h 2, C 2h 2 ] (see (5.103) in [3]). Let ( 1/2 K (3.7) v,h = inf v 2 L (Ω)), v V 2 h, V 1(v) =0 where inf V1(v) denotes the infimum with respect to all representations {v } in V 1 (v). The following important statement is similar to Corollary 2.1 in [32] or Lemma 2 in [31].

6 6 R. Aitbayev Lemma 3.4. The norms H 2 (Ω) and,h are uniformly equivalent on V h ; that is, (3.8) C 1 v H 2 (Ω) v,h C 2 v H 2 (Ω), v V h. Proof. First, we prove the second inequality in (3.8). It is nown that the Besov space B 2 2,2(Ω) coinsides, up to equivalent norms, with the Sobolev space H 2 (Ω) (see Part (b) of Theorem and (3) of in [38]). Since H 2 0 (Ω) is a closed subspace of H 2 (Ω), it is a closed subspace of B 2 2,2(Ω). We note, that functions in H 2 0 (Ω) are continuous on Ω, therefore, for any v H 2 0 (Ω), v(x) = 0 for all x Ω; in particular, the trace of v H 2 0 (Ω) is continuous. In a natural way, we extend the definition of {V } for > K. It follows from Theorem 5.1 in [16] that ( v A 2 2,2 = v 2 + ( ) ) 2 1/2 2 4 inf v z L 2 (Ω) L 2 (Ω) z V =0 is a norm on H 0 2 (Ω) equivalent to the standard norm in B2,2(Ω) 2 (see [30] for a definition of the approximation space A 2 2,2). Therefore, using the equivalence of norms H 2 (Ω) and A 2 2,2 on H 0 2 (Ω) and the fact V h H 0 2 (Ω), we get (3.9) v A 2 2,2 C v H 2 (Ω), v V h. Let us prove (3.10) v,h C v A 2 2,2, v V h. We note, (3.11) since ( K 1 v A 2 2,2 = v 2 + ( ) ) 1/ inf v z L 2 (Ω) L 2 (Ω), v V h, z V =0 inf v z L 2 (Ω) = 0, K, v V h. z V Tae any v V h and set v 0 = z 0 and v = z z 1 for 1 K, where z is the orthogonal L 2 -projection of v into V. Note, v = z K = K =0 v. Using v 0 L 2 (Ω) v L 2 (Ω), the triangle inequality, the definition of {z }, and (3.11), we obtain K =0 v 2 L 2 (Ω) = v 0 2 L 2 (Ω) + K =1 K 1 v L 2 (Ω) =0 K 1 = v L 2 (Ω) =0 z v + v z 1 2 L 2 (Ω) v z 2 L 2 (Ω) ( 2 4 inf v z L 2 (Ω) z V ) 2 4 v 2 A 2 2,2.

7 Multilevel preconditioners for OSC problems 7 Taing the infimum over V 1 (v), we get (3.10). Inequalities (3.10) and (3.9) imply the second inequality in (3.8). Let us prove the first inequality in (3.8). Tae any v V h and let {v } V 1 (v). Using the strengthened Cauchy-Schwarz inequality v v l dx C2 l /2 (h h l ) 2 v L 2 (Ω) v l L 2 (Ω), v V, v l V l, Ω (see Lemma 5.1 in [40]) and the fact that the spectral radius of matrix B = (b l ) with the entries b l = 2 l /2, 0, l K, is bounded by the maximum norm B , we get K K v 2 = ( v) 2 dx = ( v L 2 (Ω) )( v l )dx = C Ω K, l=0 C( ) Ω =0 l=0 2 l /2 (h h l ) 2 v L 2 (Ω) v l L 2 (Ω) v 2 L 2 (Ω). K, l=0 Ω v v l dx From the last estimate, using (3.2) and taing the infimum over V 1 (v), we obtain the first inequality in (3.8). To finish the proof of the lemma, we establish that,h is a norm on V h. It follows from the inequalities in (3.8) that v,h 0 for any v V h, and v,h = 0 if and only if v = 0. Let us show that cv,h = c v,h for any v V h and c R. Since the case c = 0 is trivial, assume c 0. Using the fact that the infimum over V 1 (cv) equals the infimum over V 1 (v), we obtain cv 2,h = inf K {w } V 1(cv) =0 w 2 L 2 (Ω) = inf K {v } V 1(v) =0 cv 2 = L 2 (Ω) c2 v 2,h, which implies the required identity. To prove the triangle inequality for h,, using the triangle inequality for the L 2 -norm and the Minowsi inequality, we obtain, for any v and w in V h, v + w 2,h = inf K {z } V 1(v+w) =0 inf inf K V 1(v) V 1(w) =0 z 2 inf L 2 (Ω) inf K V 1(v) V 1(w) =0 v + w 2 L 2 (Ω) ( v L 2 (Ω) + w L 2 (Ω) ) 2 ( v,h + w,h ) 2. Thus,,h is a norm on V h which is equivalent to the H 2 -norm. Remar. The result of Lemma 3.4 is analogous to those formulated in [32, Corollary 2.1] and [31, Lemma 2], where relations similar to (3.8) were proved first for Sobolev spaces with equivalent norms involving infinite number series, and then the versions for finite dimensional subspaces were obtained. Our proof of Lemma 3.4 is somewhat different since it is based on the representation (3.11).

8 8 R. Aitbayev Let (3.12) v Σ, = inf v i 2 L V 2 (Ω) 2(v),i 1/2, v V h. Lemma 3.5. (3.13) C 1 v H 2 (Ω) v Σ, C 2 v H 2 (Ω), v V h. Proof. We call nonnegative quantities A(h) and B(h) uniformly equivalent with respect to h and write A(h) B(h) if C 1 B(h) A(h) C 2 B(h). Our proof consists in establishing the following sequence of equivalence relations: (3.14) v 2 Σ, inf V 2(v),i v i 2 L 2 (Ω) v 2,h v 2 H 2 (Ω). The last equivalence relation in (3.14) is stated and proved in Lemma 3.4. Let us prove the first equivalence relation in (3.14). Tae any v V h and consider a representation {v i } V 2 (v). Using (2.10), (3.3), (3.2), and (2.9) with α = 2 and α = (0, 0), we obtain C 1 v i 2 L 2 (Ω) h 4 v i 2 L 2 (Ω) C 2 v i 2 L 2 (Ω). Summing these inequalities with respect to and i and taing the infimum over V 2 (v), we obtain the first equivalence relation in (3.14). We now prove the second equivalence relation in (3.14): (3.15) C 1 inf V 2(v),i v i 2 L 2 (Ω) v 2,h C 2 inf V 2(v) Tae any v V h. Using uniqueness of the representation,i N v = v i, v i V i, 1 i N, i=1 v i 2 L 2 (Ω), v V h. for v V, we define injection mappings V 1 (v) V 2 (v) and V 2 (v) V 1 (v) by v = v i. i The Schroeder-Bernstein theorem implies that there is a bijection V 1 (v) V 2 (v). Consider any representation {v } V 1 (v) and the representation {v i } V 2 (v) given by the bijection V 1 (v) V 2 (v). Let c i be such that v i = c i ψi. Using (2.9) with α = (0, 0), we have C 1 c 2 ih 2 v i 2 L 2 (Ω) C 2 c 2 ih 2.

9 Multilevel preconditioners for OSC problems 9 Summing the last inequalities with respect to i and using (3.6), we obtain (3.16) v i 2 v L 2 (Ω) 2 C N L 2 (Ω) 2 v i 2. L 2 (Ω) C 1 N i=1 i=1 Multiplying (3.16) by, summing for = 0, 1,..., K, taing the infimum over V 1 (v), and using (3.7), we obtain (3.17) C 1 inf v i 2 L 2 (Ω) v 2,h C 2 inf v i 2. L V 2 (Ω) 1(v) V 1(v),i Since there is a bijection V 1 (v) V 2 (v), the infimum over V 1 (v) is equal to the infimum over V 2 (v); hence, (3.17) implies (3.15). 4. Uniform spectral equivalence. In this section, we define an additive and a multiplicative OSC preconditioners and prove that they are uniformly spectrally equivalent to the OSC operator L h L h. Additive preconditioner. For 0 K and 1 i N, let Ti be a linear operator from V h into V i defined as follows: for any w V h, Ti w satisfies,i (4.1) a h (T i w, v) = a h (w, v) for all v V i. The following is our main result for the additive preconditioner. Theorem 4.1. Assume that h sufficiently small. Linear operator (4.2) T A =,i T i is selfadjoint positive definite on V h in the inner product a h (, ). Linear operator (4.3) B A = L hl h T 1 A is selfadjoint positive definite on V h in the inner product (, ) h, and (4.4) C 1 (B A v, v) h (L hl h v, v) h C 2 (B A v, v) h, v V h. (4.5) Proof. Let v 2 Σ,a h = inf V 2(v) a h (v i, v i ), v V h.,i First, let us prove the equivalence relation (4.6) C 1 a h (v, v) v 2 Σ,a h C 2 a h (v, v), v V h. Using (3.4), (3.12), and (4.5), we obtain inequalities C 1 v Σ, v Σ,ah C 2 v Σ,, v V h, which, by Lemma 3.5, imply C 1 v H 2 (Ω) v Σ,ah C 2 v H 2 (Ω), v V h.

10 10 R. Aitbayev The last inequalities and Lemma 3.1 imply (4.6). We note that the second inequality in (4.6) is one of the ey assumptions in the abstract theory of Schwarz methods (see Assumption 1 in Section 5.2 of [36]). Using (4.2), (4.1), the second inequality in (4.6), and (3.1), we obtain a h (T A v, v) Ca h (v, v) C v 2 H 2 (Ω), v V h, (see Theorem 1 in [17]). Therefore, operator T A is positive definite. Operators Ti are selfadjoint since a h (, ) is a symmetric bilinear form; hence, T A is selfadjoint (see Lemma 2 in Section 5.2 of [36]). Thus, operator T 1 A is selfadjoint positive-definite. It follows from (4.3), (2.7), and Lemma 1 in 5.2 of [36] that (B A v, v) h = a h (T 1 A v, v) = v 2 Σ,a h, v V h. The last relation along with (4.6) and (2.7), gives (4.4). Multiplicative preconditioner. Let (4.7) [ 0 N ( T M = I h Ih Ti ) ] [ K 1 ( Ih Ti ) ], =K i=1 =0 i=n where I h is the identity operator on V h. Theorem 4.2. Assume that h is sufficiently small. Linear operator T M is selfadjoint positive-definite on V h in the inner product a h (, ). Linear operator (4.8) B M = L hl h T 1 M is self-adjoint positive definite on V h in the inner product (, ) h, and (4.9) C 1 (B M v, v) h (L hl h v, v) h C 2 (B M v, v) h, v V h. Proof. Since operators {Ti } are self-adjoint on V h with respect to the inner product a h (, ), it is easy to see that T M are also a self-adjoint operator. Hence, B M is self-adjoint in the inner product (, ) h. Inequalities in (4.9) follow from Lemma 4 in 5.2 of [36] with ω = 1, the second inequality in (4.6), and Lemma 6.1 in [40] formulated for {V }. Remar. Using the multigrid terminology, we note that the multiplicative preconditioner B M corresponds to the V-cycle multigrid algorithm with the Gauss-Seidel smoother. Iterative method. Since the operator of the equation (2.6) is selfadjoint positive definite, we can use our multilevel preconditioners with the PCG algorithm to compute the OSC solution (see Algorithm in [20]). Let λ min,h and λ max,h be respectively the smallest and the largest eigenvalues of the preconditioned operator à h = M 1/2 h L hl h M 1/2 h, where M h is a preconditioner. It is well nown that the convergence rate of PCG is bounded from above by ( κ h 1)/( κ h + 1),

11 Multilevel preconditioners for OSC problems 11 where κ h = λ max,h /λ min,h is the spectral condition number of Ãh (see Theorem in [20]). It follows from theorems 4.1 and 4.2 that (4.10) κ h C 2 /C 1 < as h. The estimate (4.10) implies that it taes O( ln ɛ ) iterations of the PCG algorithm with the multilevel OSC preconditioners to approximate the solution of (2.6) with tolerance ɛ; that is, the number of iterations is bounded by a constant independent of h and K. 5. OSC matrix-vector representation. In this section, we introduce the matrix-vector representation of the OSC problem in the standard Hermite finite element basis and obtain recurrence relations for the computation of the OSC approximations and other required quantities. Representation of Hermite piecewise cubic polynomials. Let V 1 denote a vector space of Hermite piecewise cubic polynomials vanishing at t = 0 and t = 1 and corresponding to the partition {t i = i/2 } 2 is M i=0 = Let V 1 of the interval [0, 1]. The dimension of (5.1) Φ = {s 0, v 1, s 1,..., v 2 1, s 2 1, s 2 } {φ i} M i=1 be the standard basis of V 1 consisting of nodal value and nodal slope basis functions v i (t) and s i (t), respectively, defined by, for 0 i 2, (5.2) v i (t j ) = δ ij, (v i ) (t j ) = 0, 0 j 2, s i (t j ) = 0, (s i ) (t j ) = h 1 δ ij, 0 j 2, where δ ij is the Kronecer delta. We note that the basis functions v0 and v 2 corresponding to the boundary points t = 0 and t = 1 are not included in Φ. The value and the slope basis functions in Φ corresponding to an interior partition node are uniquely expressed as a linear combination of five basis functions in Φ +1 as follows: (5.3) Let (5.4) P = vi = 1 2 v+1 2i s+1 2i 1 + v+1 2i v+1 2i s+1 2i+1, s i = 1 8 v+1 2i s+1 2i sl+1 2i v+1 2i s+1 2i+1. ( ) ( 8 0, Q = 0 4 ) ( 4 1, and R = 6 1 ) be matrices corresponding to the representation (5.3). Let P 1 be a 2M M matrix obtained from Q O O... O R P O 1 O Q O (5.5) 8 O R P R (2M +2) (M +2), O... Q

12 12 R. Aitbayev where O is the 2 2 zero matrix, by removing the first and next to the last rows and columns. For v V 1, let [v] H, denote the vector representation of v in the basis Φ. It follows from (5.3), (5.4) and (5.5), that (5.6) [v] H,+1 = P 1 [v] H,, v V 1, {0, 1, 2,..., K 1}. Representation of Hermite piecewise bicubic polynomials. We note that V = V 1 V 1, where the symbol denotes a vector space tensor product. Set (5.7) ψ M (i 1)+j (x) = φ i(x 1 )φ j (x 2 ) for 1 i, j M, where basis functions φ i, for 1 i M, are defined by (5.1) and (5.2). The set Ψ = {ψj }N j=1, where N = M 2 = 4+1, is the standard basis for V in the standard ordering. It follows from (5.3) and (5.7) that a basis function in Ψ corresponding to an interior partition node is uniquely expressed as a linear combination of 25 basis functions in Ψ +1. Let [v] H, denote the vector representation of v V in the basis Ψ, and let [v] H = [v] H,K for v V h. It is obvious that (5.8) [ψ j ] H, = e j, 1 j N, where e j is the j-th standard basis vector in RN. We set (5.9) P = P 1 P 1 R 2N N, where P 1 is the one-dimensional interpolation matrix in (5.6) and the symbol now denotes the matrix tensor product. Matrix P corresponds to the piecewise bicubic Hermite interpolation in V +1, and we call P the interpolation matrix from level to level + 1. It follows from (5.6), (5.7), and (5.9) that (5.10) [v] H,+1 = P [v] H, for 0 K 1 and v V. Applying formula (5.10) recurrently, we obtain (5.11) [v] H = P K 1... P [v] H,, v V. In particular, replacing v in (5.11) by ψ j and using (5.8), we get (5.12) [ψ j ] H = P K 1... P e j, 1 j N. Matrix-vector form of the OSC problem. Assume that the set of Gauss points G h = {ξ i } N i=1 is ordered, and let, for any continuous on Ω function v(x), From (2.5), we have [v] G = [v(ξ 1 ),..., v(ξ N )] t. (5.13) (v, w) h = (h 2 /4)[w] t G[v] G, v, w V h. Let [L h ] be the OSC matrix of size N N, corresponding to the differential operator L in (1.2), with entries Lψ K j (ξ i), where i is the row index. For a continuous function g(x), let D(g) = diag (g(ξ 1 ),..., g(ξ N ))

13 be a diagonal matrix. We note that Multilevel preconditioners for OSC problems 13 [L h ] = D(a 11 )(Â ˆB) + 2D(a 12 )(Ĉ Ĉ) + D(a 22)( ˆB Â) +D(b 1 )(Ĉ ˆB) + D(b 2 )( ˆB Ĉ) + D(c)( ˆB ˆB), and the matrices Â, Ĉ, and ˆB have the following almost bloc diagonal structure: W Z O O... O Õ Õ W Z O Õ O W Z R MK MK, Õ O... W Z where W, Z, O and W, Z, Õ are, respectively, 2 2 and 2 1 blocs; O and Õ are zero matrices. The (i, j)-entries of matrices Â, Ĉ, and ˆB are φ i (η j), φ i (η j), and φ i (η j ), respectively, where {η j } MK j=1 is the set of Gauss points in interval [0, 1] corresponding to partition π h. It is easy to see that (5.14) [L h v] G = [L h ][v] H, v V h. Using (2.7), (5.13), and (5.14), we obtain, for any v and w in V h, (5.15) a h (v, w) = (L h v, L h w) h = (h 2 /4)[L h w] t G[L h v] G = (h 2 /4)[w] t H[L h ] t [L h ][v] H. Let A = (a ij ) be an N N matrix with entries (5.16) a ij = (4/h 2 )a h (ψ j, ψ i ), and let A = A K. From (5.16), using (5.15), we get (5.17) a ij = [ψ j ] t H[L h ] t [L h ][ψ i ] H. From (5.17) for = K, using (5.8), we obtain (5.18) A = [L h ] t [L h ]. Similarly, using (5.13), (5.14), the relation [f] G = [f h ] G, and (5.8), we obtain (4/h 2 )(L hf h, ψ K i ) h = (4/h 2 )(f h, L h ψ K i ) h = [f h ] t G[L h ψ K i ] G = [f] t G[L h ][ψ K i ] H = ( e K i ) t [L h ] t [f] G. Thus, the variational OSC equation (2.8) has the matrix-vector form (5.19) A [u h ] H = [L h ] t [f] G. 6. Implementation. In this section, we describe implementations of the additive and the multiplicative OSC preconditioners. Additive preconditioner. Let us describe the computation of w = B 1 A v for v V h, where B A is defined by (4.3). Let (6.1) N w = w i, where w i = Ti (L hl h ) 1 v. i=1

14 14 R. Aitbayev Using (4.3), (4.2), and (6.1), we get (6.2) w = T A (L hl h ) 1 v = (,i T i )(L hl h ) 1 v =,i w i = w. Thus, to compute w, we need to compute and sum w for = 0, 1,..., K. Using (4.1) and (2.7), we obtain, for 1 i N, (6.3) a h (w i, ψ i ) = a h (T i (L hl h ) 1 v, ψ i ) = a h ((L hl h ) 1 v, ψ i ) = (v, ψ i ) h. Let (6.4) (6.5) w i = c i ψ i, c i R, [v] = (4/h 2 )[(v, ψ 1 ) h,..., (v, ψ N ) h ] t R N. Substituting (6.4) into (6.3) and using (6.5) and (5.16), we rewrite (6.3) in the form (6.6) diag(a ) w = [v], where w = (c 1,..., c N ) t = [w ] t H, and diag(a ) = diag(a 11,..., a N N ). Let [I h ] be an N N matrix with entries ψ K j (ξ i), where i is the row index. Matrix [I h ] is nonsingular and maps [v] H to [v] G, that is, (6.7) [v] G = [I h ] [v] H, v V h. For [v] defined by (6.5), let us show (6.8) (6.9) [v] K = [I h ] t [v] G, [v] = P t [v] +1 for = K 1, K 2,..., 0, where the interpolation matrix P is defined by (5.9) and (5.5). Using (6.5), (5.13), and (6.7), we obtain (6.10) ( e j ) t [v] = (4/h 2 )(v, ψ j ) h = [ψ j ] t G[v] G = [ψ j ] t H[I h ] t [v] G, 1 j N. Relation (6.8) follows from (6.10) with = K and (5.8). Using (6.10), (5.12), and (6.8), we obtain [v] = P t... P t K 1[v] K, which implies (6.9). The multiplication by P t is carried out using the representation P t = ((P 1 ) t I )(I (P 1 ) t ), where I is the identity matrix of size M M. Matrices {A } can be precomputed using the recurrence formula (6.11) A = P t A +1 P for = K 1, K 2,..., 0, which follows from (5.17), (5.12), and (5.18). Finally, to compute w = w, that is, [w] H, we implement w +1 w +1 + P w, = 0, 1,..., K 1.

15 Multilevel preconditioners for OSC problems 15 input: K, [v] K, {diag(a )} K =0 output: [w] H v K [v] K for = K, K 1,..., 0 if ( < K) v = P t v +1 end solve diag(a ) w = v end for = 0, 1,..., K 1 w +1 w +1 + P w end [w] H w K Fig Additive preconditioning algorithm. The additive preconditioning algorithm is presented in Figure 6.1. It is easy to see that the computational cost of the additive algorithm is O(h 2 ) = O(4 K ). Multiplicative preconditioner. We now consider the computation of w = B 1 M v for v V h, where B M is defined in (4.8). Let (6.12) u = (L hl h ) 1 v. Using (4.8) and (6.12), we get which, by (4.7), implies w = B 1 M v = T M (L hl h ) 1 v = T M u, (6.13) [ 0 N ( u w = Ih Ti ) ] [ K 1 ( Ih Ti ) ] u. =K i=1 =0 i=n Let S be an ordered set of pairs (, i) with the ordering corresponding to that of factors in (6.13) from right to left. Setting y = u w, we see that u w can be computed by which is equivalent to y u; y (I h T i )y for (, i) S, (6.14) w 0; w w + T i (u w) for (, i) S. Let us develop an efficient implementation of the algorithm in (6.14). Using (4.1), (2.7), and (6.12), we obtain (6.15) a h (T i (u w), ψ i ) = a h (u w, ψ i ) = (v, ψ i ) h a h (w, ψ i ), ψ i V i. Substituting T i (u w) = c iψ i (6.16) (6.17) into (6.15) and (6.14), we get c i = g i (w)/a ii, w w + c i ψ i, where a ii is defined in (5.16), and (6.18) g i (w) = (4/h 2 ) [ (v, ψ i ) h a h (w, ψ i ) ], 1 i N.

16 16 R. Aitbayev Let g (w) = (g 1 (w),..., g N (w)) t. We note that, each time when the value of w is changed by (6.17), all entries of vector g (w) should be updated by (6.18), and such computation requires a matrix-vector product. It is more efficient to use a recurrent assignment (6.19) g (w) g (w) c i (a 1i,..., a N,i) t, which is obtained by multiplying (6.17) by ψj in the a h(, ) inner product and subtracting (v, ψj ) h from both sides of the resulting assignment. For = K 1, K 2,..., 0, let w be the value of w after implementing and let w w + T l i (u w), i = 1,..., N l, l = K, K 1,..., + 1, (6.20) g = g (w ). We note that (6.16) followed by (6.19) for i = 1,..., N is the column-oriented algorithm of solving a lower triangular linear system L w = g, where matrix L contains the lower triangular part of A and w = (c 1,..., c,n ) t. Let us show that vectors { g } can be computed by the recurrence formula (6.21) g 1 = P t 1 ( g A w ). Using (6.19), the definition of w 1, and (6.20), we obtain (6.22) g (w 1 ) = g A w. Applying (6.18), (6.5), (5.15), (5.12), we get (6.23) g (w) = [v] P t... P t K 1[L h ] t [L h ][w] H. From (6.23) with replaced by 1 and (6.9), we have g 1 (w) = P t 1 ( [v] P t... P t K 1[L h ] t [L h ][w] H ) = P t 1 g (w), which, by (6.22), implies (6.21). In the ascend phase, for = 0,..., K, an upper triangular linear system with the matrix L t is solved. The algorithm implementing the multiplicative preconditioning is given in Figure 6.2. It is easy to see that the cost of the multiplicative algorithm is O(h 2 ) = O(4 K ). 7. Numerical results. In this section, we present numerical results for solving test problems by the PCG method with the multilevel OSC preconditioners developed in this wor. For a chosen exact solution u(x) of BVP (1.1), we set f = Lu. PCG iterations are stopped when the relative residual norm, that is, the ratio of the 2-norm of the residual of equation (5.19) to the 2-norm of the right hand side, becomes less than tolerance ɛ. The spectral condition number κ h of the preconditioned OSC operator satisfies (4.10), that is, κ h < C 2 /C 1 as h 0. To demonstrate this fact numerically, we compute quantities that approximate κ h on a sequence of nested partitions using the PCG iteration parameters and present these approximations in the following tables

17 Multilevel preconditioners for OSC problems 17 input: K, [v] K, {A } K =0 output: [w] H g K [v] K for = K,..., 0 solve L w = g if ( > 0) g 1 P t 1 ( g A w ) end end for = 0,..., K if ( > 0) w w + P 1 w 1 end solve L t w = g A w w w + w end [w] H w K Fig Multiplicative preconditioning algorithm. Table 7.1 Comparison of multilevel preconditioners to solve the normal and the original OSC equations by PCG (L =, ɛ = ). Normal OSC equation Original OSC equation Additive Multiplicative Additive Multiplicative h κ h Iter. κ h Iter. κ h Iter. κ h Iter. 1/ / / / / under κ h. Under Iter., we report the numbers of iterations to reduce the relative residual norm within the specified tolerance. In the first set of experiments, we compare our preconditioners with those proposed in [9], where an additive and a multiplicative multilevel preconditioners were developed to solve the original OSC equation (2.3) for a selfadjoint L. As in [9], we tae L =, the Laplacian, and u(x) = 10x 2 1(1 x 1 )x 2 2(1 x 2 ). Since u(x) is a bicubic polynomial, it is also the solution of the OSC problem; hence, the discretization error is zero. The numerical results are presented in Table 7.1, and they indicate that, for L =, PCG with multilevel preconditioners for the original OSC equation is slightly more efficient than that for the normal OSC equation. For smaller values of h, the approximations to κ h and the numbers of iterations are relatively small and change insignificantly. In the next set of experiments, we solve the same problems as in [1], where PCG algorithm was tested with a direct solver preconditioner. The operator L in (1.2) is taen with the coefficients (7.1) a 11 (x) = e x1x2, b 1 (x) = x 2 e x1x2 + β 1 cos[π(x 1 + x 2 )], a 12 (x) = α/(1 + x 1 + x 2 ), b 2 (x) = x 1 e x1x2 + β 2 sin(2πx 1 x 2 ), a 22 (x) = e x1x2, c(x) = γ[1 + 1/(1 + x 1 + x 2 )],

18 18 R. Aitbayev Table 7.2 Approximations to the spectral condition number κ h and PCG iteration numbers for the variable coefficient L (ɛ = ). Top part multilevel preconditioner; bottom part additive preconditioner P1 P2 P3 P4 h κ h Iter. κ h Iter. κ h Iter. κ h Iter. 1/ (26) (46) (34) (68) 1/ (30) (51) (38) (75) 1/ (33) (54) (40) (81) 1/ (34) (55) (42) (84) 1/ / / / / / where α, β 1, β 2, and γ are parameters, and the exact solution of BVP (1.1) is set to u(x) = e x1+x2 x 1 x 2 (1 x 1 )(1 x 2 ). Using PCG with the multiplicative preconditioner, we solve the following four test problems corresponding to the differential operator L, which is: P1 selfadjoint negative-definite, α = β 1 = β 2 = γ = 0; P2 selfadjoint indefinite, α = β 1 = β 2 = 0 and γ = 100; P3 nonselfadjoint indefinite, β 2 = 100 and α = β 1 = γ = 0; P4 nonselfadjoint indefinite, α = 0.5, β 1 = 10, β 2 = γ = 50. In the top part of Table 7.2, we report results with the multiplicative preconditioner, and, in parentheses, we reproduce results reported in [1]. For all four problems, the numbers of iterations increase slowly with decreasing h. It taes the least number of iterations to solve the selfadjoint negative definite problem P1, and the most number of iterations to solve P2, the most indefinite problem among P1 P4. For P1, approximations to the spectral ratio κ h are much smaller than those for P2 P4, where the approximations to κ h are about the same for smaller values of h. It is interesting to note, that κ h monotonically increases for P1 and decreases for P2 P4 as h decreases. We observe that the approximations to κ h for P2 P4 are significantly larger than those for P1. The numbers of iterations for P1 and P4 indicate in favor of the multilevel multiplicative preconditioner in comparison with the direct solver preconditioner developed in [1]; although, the preconditioner in [1] produces smaller numbers of iterations for P2 and P3. Results for the additive preconditioner are presented in the bottom part of Table 7.2, and they suggest that, for the tested problems, the additive preconditioner is less efficient, in terms of numbers of iterations, than the multiplicative preconditioner. The approximations of κ h computed with the additive preconditioner are approximately 15 times larger than those computed with the multiplicative preconditioner, and the indicated difference is reflected by a larger number of iterations for the additive preconditioner. This result is intuitively expected based on nown properties of Jacobi and Gauss-Seidel smoothers for finite-difference operators. In Figure 7.1, we display plots of residual curves for h = 1/256. We observe monotone convergence only for the selfadjoint negative definite problem P1; for P2

19 Multilevel preconditioners for OSC problems 19 Fig Logarithmic plots of the relative residual norm vs. iteration number (h = 1/256). Top figure multiplicative preconditioner; bottom figure additive preconditioner P1 P2 P3 P P P P1 P Table 7.3 Approximations to the spectral condition number κ h Helmholtz equation (ɛ = ). and PCG iteration numbers for the 2 = = = = 1400 h κ h Iter. κ h Iter. κ h Iter. κ h Iter. 1/ / / / / P4, the residual curves are plotted relatively close to each other. In the last set of experiments, we solve the Helmholtz equation u + 2 u = f for several values of 2. Numerical results were obtained using the multiplicative preconditioner, and they are presented in Table 7.3. We see that the approximations to κ h change insignificantly when h is decreased for all taen values of 2. For 2 = 1000, the approximations to h are very large, however, for 2 = 1400 they are the smallest. The approximations to h are large because of small values of the smallest eigenvalue λ min,h of the preconditioned operator. In Figure 7.2, we plot the eigenvalues of the OSC matrix [L h ] with h = 1/32 (N = 4, 096) for the Helmholtz equation with 2 = We note that the eigenvalues are widely spread on the complex plane. In Figure 7.3, we plot the eigenvalues of the symmetric matrix [I h ] t [L h ], which are the eigenvalues of the OSC operator L h. We note that L h has large numbers of both positive and negative eigenvalues. Acnowledgment. The author wishes to than Prof. Bernard Bialeci for his assistance during the preparation of this paper and the referees for the valuable comments.

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