This article is cited in 2 papers

Methods and algorithms of computational mathematics and their applications

Spectral preconditioner for solving the Poisson equation

A. A. Manaev^a, V. V. Lisitsa<sup>b

a</sup> Trofimuk Institute of Petroleum Geology and Geophysics SB RAS
^b Sobolev Institute of Mathematics SB RAS

Abstract: The paper presents an approach to constructing a preconditioner for numerically solving the Poisson equation for a substantially inhomogeneous medium in application to problems of computational rock physics. The preconditioner is an operator inverse to the discrete Laplace operator, but for a simplified, layered model of the medium. In this case, the Laplace operator is inverted using spectral decomposition in one of the spatial directions and a sweep method for a series of one-dimensional problems in the second direction. This approach to constructing a preconditioner ensures that the number of iterations does not depend on the size of the problem being solved, which is confirmed by a series of numerical experiments. An important feature of the proposed approach is the use of layered models of the medium to construct the preconditioner, which increases the convergence rate of the conjugate gradient method by 10–40% compared to using a preconditioner based on inverting the Laplace operator for a homogeneous medium. In this case, the acceleration depends on the contrast of the coefficients of the original model; with increasing contrast, the efficiency of the proposed approach also increases.

Keywords: Poisson equation, Conjugate gradient method, spectral decomposition.

UDC: 519.632.4

Received: 03.03.2025
Accepted: 15.03.2025

DOI: 10.26089/NumMet.v26r208