Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

Dörfler, Willy; Findeisen, Stefan; Wieners, Christian

doi:10.1515/cmam-2016-0015

Cited by 49 publications

(55 citation statements)

References 30 publications

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“…which follows from integration by parts and the Trefftz property. Subtracting these terms from the bilinear form A, using the jump identities (4), the inequalities (9), the definition of γ in (8), and the weighted Cauchy-Schwarz inequality, we show that the form A is coercive in ||| · ||| DG norm with unit constant. Indeed, for all (w, τ ) ∈ T(T h ), we have…”

Section: Well-posednessmentioning

confidence: 99%

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

Moiola

Perugia

2017

Numer. Math.

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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. (2016, IMA J. Numer. Anal., 36, 1599-1635. 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 skeletonbased 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.1 studied in [3]. Another Trefftz discontinuous Galerkin (DG) formulation for time-dependent electromagnetic problems formulated as first-order systems has been proposed in [25] and analysed in [24] in one space dimension; it has been extended to full three-dimensional Maxwell equations in [10,11,23].We mention here that, independently of the Trefftz approach, space-time finite elements for linear wave propagation problems, originally introduced in [21] (see also [15,22]), have been used in combination with DG formulations e.g. in [7,14,35] and, more recently, in [8,16,17,27].In this paper we extend the Trefftz-DG method of [24] to initial boundary value problems for the acoustic wave equations posed on Lipschitz polytopes in arbitrary dimensions. We write the acoustic wave problem as a first-order system, as it is originally derived from the linearised Euler equations, [6, p. 14]; we consider piecewise-constant wave speed, Dirichlet, Neumann and impedance boundary conditions. The main focus of this paper is on the a priori error analysis of the Trefftz-DG scheme.The DG formulation proposed can be understood as the translation to time-domain of the Trefftz-DG formulation for the Helmholtz equation of [18], which in turn is a generalisation of the Ultra Weak Variational Formulation (UWVF) of [5]. The DG numerical fluxes are upwind in time and centred with a special jump penalisation in space. Under a suitable choice of the numerical flux coefficients, combining the proposed formulation with standard discrete spaces and complementing it with suitable volume terms, one recovers the DG formulation of [35], cf. Remark 4.2 below. The Trefftz formulation for Maxwell's equations of [10,11,23,25] corresponds to the "unpenalised" version of that one proposed here (the numerical experiments in [24, §7.5] show that the numerical error depends very mildly on the penalisation parameters).We first describe the IBVP under consideration in §2, the assumptions on the mesh in §3 and the Trefftz-DG formulation in §4. Following the thread of [18,24], in §5.2 and §5.3 we prove that the scheme is well-posed, quasi-optimal, dissipative (quantifying dissipation using the jumps of the discrete solution), and derive error estimates for some traces of the solution on the mesh skeleton. ...

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Section: Well-posednessmentioning

confidence: 99%

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

Moiola

Perugia

2017

Numer. Math.

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“…In [7], a global approximate test space Y δ ⊊ Z δ ⊂ Y is constructed by using an appropriate so-called test search space Z δ similar to [11]. Finally, the authors of [12] employ a discontinuous Galerkin approximation in space and a conforming Petrov-Galerkin approximation in time resulting in a suboptimal inf-sup constant, in particular w.r.t. time [12,Lemmata 1 and 3].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, the authors of [12] employ a discontinuous Galerkin approximation in space and a conforming Petrov-Galerkin approximation in time resulting in a suboptimal inf-sup constant, in particular w.r.t. time [12,Lemmata 1 and 3].…”

Section: Introductionmentioning

confidence: 99%

(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov--Galerkin Methods

Brunken

Smetana

Urban

2019

SIAM J. Sci. Comput.

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We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable test space and defining an optimal trial space by the application of the adjoint operator. As a result, the inf-sup constant is one in the continuous as well as in the discrete case and the computational realization is therefore easy. In particular, regarding the latter, we avoid a stabilization loop within the greedy algorithm when constructing reduced models within the framework of reduced basis methods. Several numerical experiments demonstrate the good performance of the new method.

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“…Recently, in other works, robust parallel geometric multigrid methods for parabolic problems using a discontinuous Galerkin space–time finite‐element approach have been proposed, where a special coarsening strategy is determined by a certain precise criterion. In the work of Dörfler et al, a multilevel preconditioner for the linear first‐order hyperbolic evolution systems by a discretization method combining discontinuous Galerkin in space and Petrov–Galerkin in time has been developed. In the work of Andreev, a block‐diagonal preconditioner resulting from a sparse algebraic wavelet‐in‐time transformation has been studied, where the individual spatial blocks are preconditioned by standard spatial multigrid in parallel.…”

Section: Introductionmentioning

confidence: 99%

“…Space-time finite-element methods have received more and more interest since the pioneering work of Hughes et al 1 ; see some recent works. [2][3][4][5][6][7][8][9][10][11][12][13][14][15] A common difficulty of these methods concerns the efficient solution of the related large-scale linear system of algebraic equations, which is often harder than the solution of the linear system from more conventional time-stepping methods. Recently, in other works, 9,11,12 robust parallel geometric multigrid methods for parabolic problems using a discontinuous Galerkin space-time finite-element approach have been proposed, where a special coarsening strategy is determined by a certain precise criterion.…”

Section: Introductionmentioning

confidence: 99%

Comparison of algebraic multigrid methods for an adaptive space–time finite‐element discretization of the heat equation in 3D and 4D

Steinbach

Yang

2018

Numerical Linear Algebra App

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Summary The aim of this work is to compare algebraic multigrid (AMG) preconditioned GMRES methods for solving the nonsymmetric and positive definite linear systems of algebraic equations that arise from a space–time finite‐element discretization of the heat equation in 3D and 4D space–time domains. The finite‐element discretization is based on a Galerkin–Petrov variational formulation employing piecewise linear finite elements simultaneously in space and time. We focus on a performance comparison of conventional and modern AMG methods for such finite‐element equations, as well as robustness with respect to the mesh discretization and the heat capacity constant. We discuss different coarsening and interpolation strategies in the AMG methods for coarse‐grid selection and coarse‐grid matrix construction. Further, we compare AMG performance for the space–time finite‐element discretization on both uniform and adaptive meshes consisting of tetrahedra and pentachora in 3D and 4D, respectively. The mesh adaptivity occurring in space and time is guided by a residual‐based a posteriori error estimation.

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Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

Cited by 49 publications

References 30 publications

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

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

(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov--Galerkin Methods

Comparison of algebraic multigrid methods for an adaptive space–time finite‐element discretization of the heat equation in 3D and 4D

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