Combined Preconditioning with Applications in Reservoir Simulation

TL;DR

This paper introduces a combined preconditioning framework that enhances the efficiency and robustness of solving large-scale symmetric positive definite systems, with applications demonstrated in reservoir simulation.

Contribution

The paper proposes a novel combined preconditioner framework that integrates a smoother and a preconditioner, proven to be positive definite and more effective than individual methods.

Findings

Noticeable speed-up in reservoir simulation problems.

The combined preconditioner outperforms standalone methods.

Theoretical guarantees on positive definiteness and condition number.

Abstract

We develop a simple algorithmic framework to solve large-scale symmetric positive definite linear systems. At its core, the framework relies on two components: (1) a norm-convergent iterative method (i.e. smoother) and (2) a preconditioner. The resulting preconditioner, which we refer to as a combined preconditioner, is much more robust and efficient than the iterative method and preconditioner when used in Krylov subspace methods. We prove that the combined preconditioner is positive definite and show estimates on the condition number of the preconditioned system. We combine an algebraic multigrid method and an incomplete factorization preconditioner to test the proposed framework on problems in petroleum reservoir simulation. Our numerical experiments demonstrate noticeable speed-up when we compare our combined method with the standalone algebraic multigrid method or the incomplete factorization preconditioner.