High-Order Ultra-Weak Variational Formulation for Maxwell Equations

The ultra-weak variational formulation (UWVF) is a Trefftz method for solving time-harmonic wave problems. The combination of flexible meshing and the use of plane wave basis functions makes the UWVF well-suited for solving Maxwell problems in inhomogeneous media and in general geometries that can be tens or even hundreds wavelengths. In a single mesh, the number of basis functions can be adjusted element-wise based on the element size, material properties in the element and the requested level accuracy of the solution. Previous studies have shown that the UWVF method can use from 3 to 130 plane waves per element. In this study, the use of high-dimensional plane waves basis for the UWVF is investigated. The aim is to find a basis which would allow elements with h=/τ ≃ 10 (where h is the length of the longest edge of the element and τ is the wavelength). Three methods for choosing the directions of the plane waves are compared in terms of accuracy of the UWVF approximation and the conditioning of the resulting matrix system. Results suggest that in a simple model problem the goal is achievable but requires over 500 plane wave basis functions per element for a tolerable accuracy.