A space–time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients

Imbert-Gérard, Lise-Marie; Moiola, Andrea; Stocker, Paul

doi:10.1090/mcom/3786

Cited by 6 publications

(8 citation statements)

References 26 publications

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“…Remark In Reference 42 weak Trefftz spaces have been used to treat the acoustic wave equation with smooth coefficients, we recall the quasi‐Trefftz spaces used there in (30). We recover this quasi‐Trefftz method if we replace

\Pi

in (9b) with a Taylor polynomial expansion of order

p-1

in the element center.…”

Section: The Methodsmentioning

confidence: 99%

“…To apply our framework we take

\mathcal{L}=\left(\begin{array}{cc}\nabla \cdotp & {c}^{-2}\frac{\partial }{\partial t}\\ {}\nabla & \frac{\partial }{\partial t}\end{array}\right)\kern2em u=\left(\begin{array}{c}\boldsymbol{\sigma} \\ {}v\end{array}\right)\kern0.5em

The local space‐time Trefftz space was introduced and analyzed in Reference 8, and is given by

\begin{align} 𝕋^{p} false(K false) = ({}, false(w, bold-italicτ false) \in ℙ^{p} {false(K false)}^{n + 1} false| \begin{matrix} subarray \\ array \nabla w + \partial_{t} τ = 0 \\ array \nabla \cdot τ + c^{- 2} \partial_{t} w = 0 \end{matrix}) \end{align}

We consider the space‐time DG‐scheme used in References 8,23,42,52:

\begin{align} Find false(v_{h p}, {bold-italicσ}_{h p} false) \in {false(V^{p} false(𝒯_{h} false) false)}^{d + 1} s.t. \\ a_{h} false(v_{h p}, {bold-italicσ}_{h p}; w, bold-italicτ false) \end{align}

…”

Section: Applicationsmentioning

confidence: 99%

“…The construction of a basis for (25) with smooth wavespeed is non‐viable. A space with similar properties, for which a basis can be constructed, has been introduced in Reference 42—the quasi‐Trefftz space for an element

K \in 𝒯_{h}

is given by

\begin{align} {ℚ 𝕋}^{p} false(K false) : = ({}, false(w, bold-italicτ false) \in ℙ^{p} {false(K false)}^{n + 1} (|, \begin{matrix} subarray \\ array D^{i} (\nabla w + \partial_{t} τ) (x_{K}, t_{K}) = 0 \\ array D^{i} (\nabla \cdot τ + c {(x)}^{- 2} \partial_{t} w) (x_{K}, t_{K}) = 0 \end{matrix}) \forall bold-italici \in ℕ_{0}^{n + 1}, false| bold-italici false| \leq p prefix- 1) . \end{align}

where we use the notation

\boldsymbol{i}=\left({\boldsymbol{i}}_{\mathbf{x}},{i}_t\right)=\left({i}_{x_1},\dots, {i}_{x_n},{i}_t\right)\in {\mathbb{N}}_0^{n+1}

for integer non‐negative multi‐indices and

D^{bold-italici} f<...

…”

Section: Applicationsmentioning

confidence: 99%

“…Remark The volume term in (27) can be omitted for the case of homogeneous media when using Trefftz or embedded Trefftz DG methods. In the inhomogeneous case, in Reference 42 the quasi‐Trefftz methods also require the volume terms for stability, however, even in the case of smooth coefficients, these terms can still be omitted when using embedded Trefftz DG method as described in Section 4.6. Note that in References 42 and 52 there appear additional Galerkin‐least squares terms needed for the analysis, however, already in Reference 42 it was observed that they do not seem to play a significant role in the numerical examples, which is why they are neglected here.…”

Section: Applicationsmentioning

confidence: 99%

“…This can be circumvented by weakening the requirements of the Trefftz space, providing again a sufficiently large basis. In Reference 42 a quasi‐Trefftz DG method is introduced and analyzed for the acoustic wave equation with smooth coefficients. Plane waves have been generalized to work with a DG method for the Helmholtz equation with smooth coefficient in Reference 43.…”

Section: Introductionmentioning

confidence: 99%

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Embedded Trefftz discontinuous Galerkin methods

Lehrenfeld

Stocker

2023

Numerical Meth Engineering

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In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using discontinuous shape functions that are chosen to be elementwise in the kernel of the corresponding differential operator. We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz‐type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non‐constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. Compared to standard discontinuous Galerkin methods we observe a severe reduction of the globally coupled unknowns in all considered cases, reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves.

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