Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations

TL;DR

This paper presents an efficient implementation of a high-order discontinuous Galerkin finite element method for solving Maxwell's equations, leveraging recurrence relations derived via computer algebra to optimize performance.

Contribution

It introduces a novel implementation technique that exploits recurrence properties and tensor-product structures, enhancing computational efficiency for Maxwell's equations.

Findings

Achieves energy-conserving discretization of Maxwell's equations.

Utilizes computer algebra to derive recurrence relations for shape functions.

Demonstrates improved efficiency in finite element computations.

Abstract

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 approximations of the electric and magnetic field. Special emphasis is placed on an efficient implementation which is achieved by taking advantage of recurrence properties and the tensor-product structure of the chosen shape functions. These recurrences have been derived symbolically with computer algebra methods reminiscent of the holonomic systems approach.