Space-time geometric multigrid method for nonlinear advection–diffusion problems

Abstract

Multigrid methods, algebraic or geometric, commonly suffer from high frequency residuals after prolongation. This paper develops a stable approach to remove high frequency residuals for geometric multigrid methods for solving nonlinear advection–diffusion problems with degenerate coefficients. Here, a local problem is treated by optimization on subdomains with mesh refinements. Newton's method is utilized in the procedure and the iteration is completed when the residual in the subdomain is reduced to the given magnitude, usually set to be the average of residuals in the non-high-frequency domains. An oversampling technique is employed to further improve the stability by providing a definite flow path in regions where coefficients have high contrast and complex structures. Removing high frequency residuals before continuing the global Newton iteration improves global convergence behavior.

Keywords:

• Local residual minimization
• smoothing
• geometric multigrid
• advection–diffusion problem
• mixed finite element method

2020 Mathematics Subject Classifications:

• 35D30
• 86-08
• 86-10
• 65M55
• 65M60

Disclosure statement

No potential conflict of interest was reported by the author(s).