A subspace method based on the Neumann series for the solution of parametric linear systems

Abstract

In this work, a subspace method is proposed for efficient solution of parametric linear systems with a symmetric and positive definite coefficient matrix of the form $I-A\left(\sigma \right)$. The motivation is to use the method for solution of linear systems appearing when solving parameter dependent elliptic PDEs using the finite element method (FEM). In the proposed method, one first computes a method subspace and then uses it to approximately solve the linear system for any parameter vector. The method subspace is designed in such a way that it contains the $j+1$-term truncated Neumann series approximation of the solution to desired accuracy for any admissible parameter vector. This allows us to use the best approximation property of subspace methods to show that the subspace solution is at least as accurate as the truncated Neumann series approximation. The performance of the method is demonstrated by numerical examples with the parametric diffusion equation. In these examples, the method yields much smaller errors than anticipated by the Neumann series based error analysis. We study this phenomenon in some special cases.