A non-overlapping domain decomposition method for incompressible Stokes equations with continuous pressure

TL;DR

This paper introduces a non-overlapping domain decomposition method for solving incompressible Stokes equations, utilizing continuous pressure spaces and Lagrange multipliers, with proven efficiency and convergence demonstrated through numerical experiments.

Contribution

The paper presents a novel domain decomposition algorithm that enforces pressure continuity without Lagrange multipliers and proves its condition number bounds are independent of subdomain count.

Findings

Condition number bound is independent of subdomain number.

Algorithm converges efficiently in numerical experiments.

Preconditioned conjugate gradient method effectively solves the reduced system.

Abstract

A non-overlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain domain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semi-definite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments demonstrate the convergence rate of the proposed algorithm.