# Efficient high-order Maxwell solvers via discrete space splittings

## Dr. Sabine Zaglmayr

**Dec. 21, 2009, 8:30 a.m. T 041**

In order to guarantee unique solvability for magnetostatic problems, additional constraints, so called gauging conditions, have to be imposed. In particular, we consider the Coulomb gauge, which enforces orthogonality of the magnetic vector potential to gradient fields. Two realizations, one based on a Lagrange multiplier formulation, and a second using a penalization strategy, will be discussed.

By a careful construction of high order finite element spaces, the Coulomb gauge can be realized by a two step strategy: First, a reduced magnetostatic problem, resulting from the elimination of a part of the high order basis functions, is solved. In a second step, the orthogonality to gradients is restored by postprocessing.

Both subproblems are substantially smaller and better conditioned than the original problem. Therefore, the overall performance of the solver is improved considerably. The efficiency of the two-step approach is also illustrated by numerical examples.