Solution Adaptive Finite Element Discretizations for the Maxwell Equations

Description

The accuracy and efficiency of numerical algorithms for the solution of the Maxwell equations can be significantly improved using adaptive finite element methods. In particular, near sharp corners and on non-convex domains the solution has singularities which can be captured more efficiently with a solution adaptive mesh. A key element in any adaptive scheme is the control of the adaptation process. In this presentation we will discuss an implicit a posteriori error estimation technique for the adaptive solution of the

Maxwell equations on three-dimensional domains using Nédélec edge finite elements. On each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis. The local error distribution is then used to generate a new mesh to obtain a more accurate numerical solution. A nice feature of the implicit a posteriori error estimates is that they do not contain unknown coefficients, as frequently occurs with residual based error estimators and improves the reliability of the adaptive algorithm. After the definition of the main algorithm we will first discuss some of its theoretical properties. In particular, we will show that the discrete bilinear form of the local problems satisfies an inf-sup condition which ensures the well posedness of the error equations. Also, the relation of the estimated error to the true error will be discussed. In the second part of this presentation the performance of the method is demonstrated on various problems, including non-convex domains with non-smooth boundaries. The numerical results show that the estimated error, computed by the implicit a posteriori error estimation technique, correlates well with the actual error.

Period	28 Mar 2007
Held at	Centre for Analysis, Scientific computing and Applications (CASA), Netherlands
Degree of Recognition	National