Matrix decomposition algorithms for elliptic boundary value problems: a survey

Abstract

We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can be employed in an MDA with a resulting computational cost of O(N 2 log N) on an N × N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two-point boundary value problems in one space dimension. In many important cases, these eigensystems are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation, finite element Galerkin and spectral methods, and the method

A parallel radix-4 block cyclic reduction algorithm

“…, f n1 denote the exact right-hand side vector blocks, f i,ε be the floating point counterpart of f i and ε ≥ 0 be selected in such a way that i . Substituting the spectral norm estimates (33) and (34) into the formula (14) gives the following error estimates for the reduction stage…”

Matrix decomposition algorithms for arbitrary order C0 tensor product finite element systems

“…This work is a substantial generalization of the case k = 2 which is considered in [2]; see also [3]. As indicated in [1], there is very little in the literature on the use of MDAs in FEG methods. The case k = 1 is described in [1] for Poisson problems, and in [4], MDAs are developed for problems arising in fluid dynamics, linear elasticity and electromagnetics.…”

“…In [1], a comprehensive survey is given of fast direct methods, called matrix decomposition algorithms (MDAs), for solving certain systems of linear algebraic equations of the form (1.1) where A 1 and B 1 are M 1 × M 1 matrices, A 2 and B 2 are M 2 × M 2 and ⊗ denotes the matrix tensor product. Such systems arise in various commonly used techniques, such as finite difference, spline collocation and spectral methods, for solving Poisson's equation −∆u = f (x, y) (x, y) ∈ Ω, (1.2) where Ω = (0, 1)×(0, 1), the unit square, with boundary ∂Ω, and ∆ denotes the Laplace operator.…”

A Kansa-RBF method for Poisson problems in annular domains

“…For any choice of RBF, for an appropriate choice of collocation points, such discretizations lead to linear systems in which the coefficient matrices possess block circulant structures. These systems can be solved efficiently using Matrix Decomposition Algorithms (MDAs) [1] with Fast Fourier Transforms (FFTs). Such MDAs have been used in the past in various applications of the Method of Fundamental Solutions to boundary value problems in geometries possessing radial symmetry, see e.g., [6,7], as well as RBF approximations and their derivatives in circular domains in [8], see also [4].…”