Computer algebra meets finite elements: An efficient implementation for maxwells equations

We consider the numerical discretization of the time-domain Maxwell’s equations with an energy-conserving discontinuous Galerkin finite element formulation. This particular formulation allows for higher order approxi-mations of the electric and magnetic field. Special emphasis is placed on an efficient implementation which is achieved by taking advantage of re-currence properties and the tensor-product structure of the chosen shape functions. These recurrences have been derived symbolically with com-puter algebra methods reminiscent of the holonomic systems approach.