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A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

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Abstract

We introduce a space–time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one-dimensional scheme of Kretzschmar et al. (IMA J Numer Anal 36:1599–1635, 2016). Test and trial discrete functions are space–time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeleton-based and mesh-independent norms. The space–time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on “tent-pitched” meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order h-convergence bounds.

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Notes

  1. Recall that if \({{\mathbf {F}}}\in H(\mathrm{div};D)\), \(D=D_1\cup D_2\cup \Sigma \subset \mathbb {R}^d\), \(d\in \mathbb {N}\), where \(D,D_1,D_2\) are Lipschitz domains with outward-pointing unit vectors \({{\mathbf {n}}}, {{\mathbf {n}}}_1,{{\mathbf {n}}}_2\), respectively and \(\Sigma \) is a Lipschitz hypersurface separating \(D_1\) and \(D_2\), then, by the divergence theorem and the well-definiteness of the normal traces in \(H(\mathrm{div};D)\) [2, eq. (2.6)],

    $$\begin{aligned} \int _\Sigma ({{\mathbf {F}}}_{|_{D_1}}\cdot {{\mathbf {n}}}_1+ {{\mathbf {F}}}_{|_{D_2}}\cdot {{\mathbf {n}}}_2)\,\mathrm {d}S&= \int _{\partial D_1}{{\mathbf {F}}}_{|_{D_1}}\cdot {{\mathbf {n}}}_1\,\mathrm {d}S + \int _{\partial D_2}{{\mathbf {F}}}_{|_{D_2}}\cdot {{\mathbf {n}}}_2\,\mathrm {d}S - \int _{\partial D}{{\mathbf {F}}}\cdot {{\mathbf {n}}}\,\mathrm {d}S\\&= \int _{D_1} \nabla \cdot {{\mathbf {F}}}_{|_{D_1}}\,\mathrm {d}V + \int _{D_2} \nabla \cdot {{\mathbf {F}}}_{|_{D_2}}\,\mathrm {d}V - \int _{D} \nabla \cdot {{\mathbf {F}}}\,\mathrm {d}V =0. \end{aligned}$$

  2. We recall that a set \(A\subset \mathbb {R}^N\) is called star-shaped with respect to a subset \(B\subset A\) if for all \({{\mathbf {a}}}\in A\) and \({{\mathbf {b}}}\in B\) the line segment with endpoints \({{\mathbf {a}}}\) and \({{\mathbf {b}}}\) is contained in A. In particular, a convex set is star-shaped with respect to any of its subsets.

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Acknowledgements

The authors are grateful to Blanca Ayuso de Dios, Thomas Hagstrom, Joachim Schöberl and Endre Süli for stimulating discussions in relation to this work.

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Author notes

  1. Andrea Moiola

    Present address: Department of Mathematics, University of Pavia, 27100, Pavia, Italy

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading, RG6 6AX, UK

    Andrea Moiola

  2. Faculty of Mathematics, University of Vienna, 1090, Vienna, Austria

    Ilaria Perugia

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Correspondence to Andrea Moiola.

Additional information

I. Perugia has been funded by the Vienna Science and Technology Fund (WWTF) through the project MA14-006, and by the Austrian Science Fund (FWF) through the projects P 29197-N32 and F 65.

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Moiola, A., Perugia, I. A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math. 138, 389–435 (2018). https://doi.org/10.1007/s00211-017-0910-x

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