A new block diagonal preconditioner for a class of $3\times 3$ block saddle point problems

TL;DR

This paper introduces a novel block diagonal preconditioner designed for $3\times 3$ block saddle point problems, demonstrating improved eigenvalue bounds and enhanced convergence in GMRES for Maxwell equations.

Contribution

The paper proposes a new block diagonal preconditioner specifically for $3\times 3$ saddle point problems, with theoretical eigenvalue bounds and practical acceleration of iterative solvers.

Findings

Eigenvalue bounds of the preconditioned matrix are established.

The preconditioner accelerates GMRES convergence.

Numerical experiments confirm the effectiveness of the preconditioner.

Abstract

We study the performance of a new block preconditioner for a class of $3\times3$ block saddle point problems which arise from finite element methods for solving time-dependent Maxwell equations and some other practical problems. We also estimate the lower and upper bounds of eigenvalues of the preconditioned matrix. \cred{Finally, we examine our new preconditioner to accelerate the convergence speed of the GMRES method which shows the effectiveness of the preconditioner.