Monolithic two-level Schwarz preconditioner for Biot's consolidation model in two space dimensions

TL;DR

This paper develops and analyzes a monolithic two-level Schwarz preconditioner tailored for efficiently solving the Biot consolidation model in two dimensions, demonstrating its effectiveness through theoretical analysis and numerical tests.

Contribution

It introduces a novel monolithic overlapping domain decomposition method for the Biot problem, with a rigorous convergence analysis and validation through numerical experiments.

Findings

The preconditioner achieves uniform convergence rates.

Numerical results confirm theoretical predictions.

Effective for two-dimensional Biot model problems.

Abstract

This paper addresses the construction and analysis of a class of domain decomposition methods for the iterative solution of the quasi-static Biot problem in three-field formulation. The considered discrete model arises from time discretization by the implicit Euler method and space discretization by a family of strongly mass-conserving methods exploiting $H^{div}$-conforming approximations of the solid displacement and fluid flux fields. For the resulting saddle-point problem, we construct monolithic overlapping domain decomposition (DD) methods whose analysis relies on a transformation into an equivalent symmetric positive definite system and on stable decompositions of the involved finite element spaces under proper problem-dependent norms. Numerical results on two-dimensional test problems are in accordance with the provided theoretical uniform convergence estimates for the two-level multiplicative Schwarz method.