Geometric multigrid with multiplicative Schwarz smoothers for eddy-current and Maxwell’s equations in deal.II

Abstract

In this work, we consider the design of a geometric multigrid method with multiplicative Schwarz smoothers for the eddy-current problem and the time-harmonic Maxwell equations. The main purpose is to show numerically that a straightforward application works for the former problem, but not for the latter. The well-known key is a special decomposition of the function spaces within the multigrid algorithm. The failures and performance are shown with the help of a numerical test, implemented in the modern finite element library deal.II, including a github link to the eddy-current implementation.

Keywords

• Time-harmonic Maxwell
• Eddy-current problem
• Finite elements
• Geometric multigrid
• Multiplicative Schwarz smoother