Dept. of Math./CMA Univ. of Oslo Pure Mathematics No 5 ISSN 0806–2439 February 2009 Foundations of finite element methods for wave equations of Maxwell type * Snorre H. Christiansen †‡ Abstract The first part of the paper is an overview of the theory of approx- imation of wave equations by Galerkin methods. It treats convergence theory for linear second order evolution equations and includes studies of consistency and eigenvalue approximation. We emphasize differential operators, such as the curl, which have large kernels and use L 2 stable interpolators preserving them. The second part is devoted to a frame- work for the construction of finite element spaces of differential forms on cellular complexes. Material on homological and tensor algebra as well as differential and discrete geometry is included. Whitney forms, their duals, their high order versions, their tensor products and their hp-versions all fit. Introduction The Yee scheme [62] is a very efficient finite difference scheme for sim- ulating the initial/boundary value problem for Maxwell’s equations. It is used in many industrial codes for problems ranging from antenna de- sign, to electromagnetic compatibility and to medical imaging. However it is only second order accurate, it treats boundary conditions in a rough way (stair-casing) and it is not clear how it should be formulated for anisotropic materials. All three problems can be addressed in the finite element framework. For electromagnetics, mixed finite elements [53] [49] [50] [15], have been found to give the best results. Finite elements come with a natural way of deriving error estimates. Stability of the method is linked to energy conservation, which is almost automatic for variational Galerkin methods. Charge conservation is also ensured weakly by these methods. Through stability, the convergence of the method is reduced to the problem of estimating errors of best approximation on the finite element space. They are usually obtained from interpolation operators. ∗ to appear in “Applied Wave Mathematics - Selected Topics in Solids, Fluids, and Math- ematical Methods” edited by Ewald Quak and Tarmo Soomere, Springer Verlag. † CMA, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway. ‡ This work, conducted as part of the award “Numerical analysis and simulations of geo- metric wave equations” made under the European Heads of Research Councils and European Science Foundation EURYI (European Young Investigator) Awards scheme, was supported by funds from the Participating Organizations of EURYI and the EC Sixth Framework Program. 1