Dr. Joachim Schöberl
Johann Radon Institute for Computational and Applied Mathematics (RICAM) Linz, Austria
High Order Nédélec Finite Elements: Construction, Preconditioning, and Eigenvalue problems
Abstract: The goal of the presented work is the efficient computation of Maxwell boundary and eigenvalue problems using high order
finite elements.
In the first part of the talk we discuss the realization of arbitrary
order hierarchic shape functions for common element geometries. The
basic strategy is due to Ainsworth and Coyle, the shape functions are
constructed block-wise as low-order Nédélec, higher-order
edge-based, face-based (only in 3D) and element-based shape functions.
Our modified is shape functions provide localized complete sequence
property in the following sense:
edge-based shape functions are gradients of
edge-based shape functions. The gradients of
face-based shape functions are
face-based ones and the gradients of
element-based shape functions are
element-based ones.
One advantage of this construction is that simple block-diagonal preconditioning gets efficient.
Our realization of the shapes allows the local elimination of
gradients. This implies a saving of degrees of freedom for
magnetostatic boundary value problems. Moreover, the curl-curl system
gets better conditioned in the reduced space. A further advantage is
that the implementation of the discrete gradient-operator from
to
can be done without any additional computational costs.
In the second part, we demonstrate an inexact version of the inverse
iteration for computing the few lowest eigen-pairs of the Maxwell
eigenvalue problem. It requires a good preconditioner for the
curl-curl system and an approximate projection into the discrete
divergence-free sub-space in order to avoid computing zero eigenvalues
due to the large known nullspace of the curl-curl matrix. The
approximate projection is implemented by one application of the
-preconditioner.
Numerical examples demonstrating the performance of the algorithm for 3D Maxwell boundary and eigenvalue problems are presented.
| Zeit: | Freitag, 8. Juli, 2005, 14.15 (Kaffee/Tee um 15.30) |
| Ort: | FU Berlin, Arnimallee 2-6, Raum 032 im EG |