# A Space-Time Parallel Multigrid Method for the Transient Eddy Current Equation

## Martin Schwalsberger

**Oct. 30, 2018, 2:30 p.m. S2 054**

In this thesis, we solve a model problem for the Eddy-current equation parallel in time and space. Throughout the thesis we discuss various aspects of our algorithm and gather knowledge about its application in more general cases. The simple version of this algorithm uses multigrid in time to solve any parabolic problem, while the more complex version uses multigrid in space and time. Therefore solving our model problem, which approximates an induction furnace, with the simple version is very straightforward and works as expected. On the contrary new concepts are required for the space-time multigrid version to deal with more general problems. This leads us to a generalized coarsening rule, which we verify for the heat equation by numerical experiments.

Our first attempts to solve the model problem with the space-time multigrid fail. A detailed analysis involving local Fourier analysis and numerical experiments, identifies the kernel of the curl operator as the root of the problem. We proceed to theoreti-cally derive an improved smoother to remedy this problem. Said smoother relies on an auxiliary nodal problem, which can be solved very efficiently. The concept for this smoother can be applied to several parabolic problems, where the spatial operator features a non trivial kernel, for example the time dependent grad-div equation.

Our main contributions are the generalized coarsening rule, which makes solving prac-tical problems more feasible, and the hybrid smoother, which enables coarsening in space. Both of these concepts could also be beneficial for related algorithms.