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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.

qtrefftz Wave propagation in inhomogeneous material using quasi-Trefftz DG