# A quasi-Trefftz space for a second order time-harmonic Maxwell's equation

## TL;DR

This paper introduces a novel local Taylor-based polynomial quasi-Trefftz space tailored for electromagnetic wave propagation in inhomogeneous media, enabling explicit construction and dimension analysis for Maxwell's equations.

## Contribution

It is the first to develop and analyze a quasi-Trefftz space for a system of PDEs, specifically for second order Maxwell's equations with variable coefficients.

## Key findings

- Explicit dimension of the quasi-Trefftz space derived
- Procedure for constructing quasi-Trefftz functions established
- Application to electromagnetic wave propagation in inhomogeneous media

## Abstract

Quasi-Trefftz methods are a family of Discontinuous Galerkin methods relying on equation-dependent function spaces. This work is the first study of the notion of local Taylor-based polynomial quasi-Trefftz space for a system of Partial Differential Equations (PDEs). These discrete spaces are introduced here for electro-magnetic wave propagation in inhomogeneous media, governed by a second order formulation of Maxwell's equation with variable coefficients. Thanks to an adequate Helmholtz decomposition for spaces of homogeneous polynomial vector fields, the outcome is the explicit dimension of the proposed quasi-Trefftz space as well as a procedure to construct quasi-Trefftz functions.