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

Abstract

Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise constant. We introduce a "quasi-Trefftz" discontinuous Galerkin method for the discretisation of the acoustic wave equation with piecewise-smooth wavespeed: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and highorder convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.

Embedded Trefftz discontinuous Galerkin methods

“…The penalization parameters in (24) are chosen as α = β = 0.5, as in [43]. The considered spacetime meshes, each is made up from a simplicial unstructured mesh of the spatial domain and a tensor product mesh in time.…”

“…Remark 7 (Quasi-Trefftz). In [24] weak Trefftz spaces have been used to treat the acoustic wave equation with smooth coefficients, we recall the quasi-Trefftz spaces used there in (27). We recover this quasi-Trefftz method if we replace Π in (9b) with a Taylor polynomial expansion of order p − 1 in the element center.…”

“…The construction of a basis for (22) with smooth wavespeed is non-viable. A space with similar properties, for which a basis can be constructed, has been introduced in [24] -the quasi-Trefftz space for an element K ∈ T h is given by…”

“…We construct the weak Trefftz embedding as described in Sections 2.3 and 3.3, using the following condition for the weak Trefftz space L(σ, v), (τ , w) = 0, ∀(τ , w) ∈ (V p−1 (T h )) d+1 . The volume term in (24) can be dropped when using the embedded Trefftz DG method, which we also did in the numerical examples in this section, cf. Remark 10.…”

“…This can be circumvented by weakening the requirements of the Trefftz space, providing again a sufficiently large basis. In [24] 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 [23].…”