A Mass Conserving Mixed hp-FEM Scheme for Stokes Flow. Part III: Implementation and Preconditioning

TL;DR

This paper develops and analyzes a block-diagonal preconditioner for a high-order mixed finite element method for Stokes flow, ensuring stability and efficient solution of the resulting linear systems.

Contribution

It introduces a preconditioning strategy for the Schur complement system that is stable across mesh sizes and polynomial degrees, with proven eigenvalue bounds.

Findings

Eigenvalue bounds are independent of mesh size h.

Preconditioner performance is robust for high polynomial degrees p.

Numerical results confirm theoretical stability and efficiency.

Abstract

This is the third part in a series on a mass conserving, high order, mixed finite element method for Stokes flow. In this part, we study a block-diagonal preconditioner for the indefinite Schur complement system arising from the discretization of the Stokes equations using these elements. The underlying finite element method is uniformly stable in both the mesh size h and polynomial order p, and we prove bounds on the eigenvalues of the preconditioned system which are independent of h and grow modestly in p. The analysis relates the Schur complement system to an appropriate variational setting with subspaces for which exact sequence properties and inf-sup stability hold. Several numerical examples demonstrate agreement with the theoretical results.