Efficient Algorithms For Solving A Fourth Order Equation With The Spectral-Galerkin Method

. We show that one can derive an O(N 3 ) spectral-Galerkin method for fourth order (biharmonic type) elliptic equations based on the use of Chebyshev polynomials. The use of Chebyshev polynomials provides a fast transform between physical and spectral space which is advantageous when a sequence of problems must be solved e.g., as part of a nonlinear iteration. This improves the result of Shen [9] which reported an O(N 4 ) algorithm inferior to the O(N 3 ) method developed earlier [8] based on Legendre polynomials, but less practical in the case of multiple problems. We further compare our method with an improved implementation of the Legendre-Galerkin method based on the same approach. Key words. spectral-Galerkin method, Chebyshev polynomial, Legendre polynomial, Helmholtz equation, biharmonic equation, direct solver, iterative solver. AMS(MOS) subject classification. 65N35, 65N22, 65F05, 35J05. 1. Introduction. In his recent work [8] and [9] Shen develops a class of spectral..

On The Spectra Of Sums Of Orthogonal Projections With Applications To Parallel Computing

. Many parallel iterative algorithms for solving symmetric, positive definite problems proceed by solving in each iteration, a number of independent systems on subspaces. The convergence of such methods is determined by the spectrum of the sums of orthogonal projections on those subspaces, while the convergence of a related sequential method is determined by the spectrum of the product of complementary projections. We study spectral properties of sums of orthogonal projections and in the case of two projections, characterize the spectrum of the sum completely in terms of the spectrum of the product. AMSMOS: 65N30 65J10 35J20 15A18 Keywords: Orthogonal Projections, Parallel Computing, Domain Decomposition, Grid Refinement, Schwarz Alternating Method. 1. Introduction. Recently there has been a strong revival of the interest in domain decomposition algorithms for elliptic problems; cf. e.g. Glowinski et al. [12], and Chan et al. [4]. A classical algorithm of this kind is the Schwarz alte..

Two Different Data-Parallel Implementations of the BLAS

Massively parallel computer systems, having thousands of identical processors operating in SIMD mode, hold the promise of delivering cost effective computing alternatives for many important problems in scientific computing. Computational linear algebra is of fundamental importance to a large class of compute intensive algorithms. This paper discusses the implementation and performance of the computational BLAS kernels in a data-parallel setting. Two different programming languages are compared and several compiler issues are discussed. Keywords: Data-parallel, Fortran 90, SIMD, BLAS, LAPACK 1 Data-parallel programming This paper discusses the data-parallel programming model applied to efficient implementation of basic computational kernels in linear algebra. This programming model has gained acceptance among computational scientists by way of the proposed array extensions to the Fortran language. These extensions constitute an important part of the new Fortran 90 standard (F90) [11]...

Parallel Domain Decomposition And Iterative Refinement Algorithms

this paper orthogonality and symmetry always refer to this inner product. The problem is discretized by finite elements in the customary fashion; cf. Ciarlet [8] and we define our discrete problem and the corresponding approximate solution u h 2