Solution Techniques for the Stokes System: A priori and a posteriori modifications, resilient algorithms

TL;DR

This paper introduces modifications to finite element methods for the Stokes system that enhance physical property preservation, improve approximation quality, and develop resilient algorithms for high-performance computing environments.

Contribution

It proposes operator modifications instead of space enrichment, local correction techniques for energy and mass conservation, and resilient algorithms for fault tolerance in supercomputing.

Findings

Enhanced energy representation and reduced pollution effects.

Achieved local mass conservation with a posteriori flux correction.

Developed resilient algorithms for fault tolerance in large-scale parallel computations.

Abstract

This article proposes modifications to standard low order finite element approximations of the Stokes system with the goal of improving both the approximation quality and the parallel algebraic solution process. Different from standard finite element techniques, we do not modify or enrich the approximation spaces but modify the operator itself to ensure fundamental physical properties such as mass and energy conservation. Special local a~priori correction techniques at re-entrant corners lead to an improved representation of the energy in the discrete system and can suppress the global pollution effect. Local mass conservation can be achieved by an a~posteriori correction to the finite element flux. This avoids artifacts in coupled multi-physics transport problems. Finally, hardware failures in large supercomputers may lead to a loss of data in solution subdomains. Within parallel multigrid, this can be compensated by the accelerated solution of local subproblems. These resilient algorithms will gain importance on future extreme scale computing systems.