Joachim Schöberl, Johannes Kepler Univeristy, Linz
Abstract. The Maxwell Equations describe electromagnetic
field phenomena. We are interested in the time harmonic setting leading
to variational problems in the function space H(curl). Proper finite elements
due to Nedelec. The lowest order member of his family is the famous edge-element.
Since many applications are real 3D problems, the arising matrix equations
are usually huge. Thus, fast solvers are very important. Two approaches
to multigrid methods (including analysis) are due to Hiptmair, and Arnold-Falk-Winther.
In this talk, we present a theoretical tool, namely a Clement type quasi-interpolation
operator for H(curl) allowing us to simplify and extend both multigrid
theories. We discuss also extensions to higher order elements, and to anisotropic
meshes. In many real life problems the coarsest mesh is already too fine
for a direct solver. For such cases, algebraic multigrid provides an attractive
alternative. We present a commuting AMG for H(curl) systems.