Fast multilevel solvers for a class of discrete fourth order parabolic problems

TL;DR

This paper develops fast, robust multilevel iterative solvers for discretized fourth order parabolic equations, demonstrating their efficiency and stability through eigenvalue analysis and numerical experiments.

Contribution

It introduces novel preconditioners using lumped mass matrices and analyzes their effectiveness with multigrid algorithms for fourth order parabolic problems.

Findings

Preconditioners are highly efficient and robust across parameters.

Eigenvalue analysis confirms rapid convergence of GMRes.

Multigrid methods with different smoothers perform effectively.

Abstract

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.