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Quasi-Trefftz DG for the wave equation

We consider the wave operator

\[\begin{align*} \begin{split} (\square_G f)(\mathbf{x},t):= \Delta f(\mathbf{x},t)-G(\mathbf{x})\partial_t^2 f(\mathbf{x},t). \end{split} \end{align*}\]

with smooth coefficient $G(\mathbf{x})$. Constructing a basis for a traditional Trefftz space (i.e. a space of functions with $\square_G f=0$) is not possible. The crucial idea is that we want to relax the Trefftz porperty to

\[\square_G f=\mathcal{O}(\|(\mathbf{x},t)-(\mathbf{x}_K,t_K)\|^q),\]

with respect to the center of a mesh element $K$ and up to some $q$. This leads to the definition of a new quasi-Trefftz space: For an element $K$ in a space-time mesh let

\[\begin{align*} \begin{split} \mathbb{T}^p(K):=\big\{ f\in\mathbb{P}^p(K) \ \mid&\ D^{i}\square_G f(\mathbf{x}_K,t_K)=0,\\ &\forall i\in \mathbb{N}^{n+1}_0, |i|<p-1 \big\}, \end{split} \end{align*}\]

For this space we are able to construct a basis. We then introduce a space-time DG method with test and trial functions that are locally quasi-Trefftz. The example below shows an acoustic wave propagating through a material with $G(x,y)=y+1$ and homogeneous Neumann boundary conditions.