Block Jacobi Relaxation for Plane Wave Discontinuous Galerkin Methods

Betcke, Timo; Gander, Martin J.; Phillips, Joel

doi:10.1007/978-3-319-05789-7_55

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…When a PUM and a standard FEM discrete spaces are combined, e.g. using formulation (13), the method obtained is termed generalised finite element method (GFEM); e.g. [108] employs high-order tensor-product polynomials summed to products of plane waves and bilinear functions.…”

Section: Non-trefftz Methods With Oscillatory Basis Functionsmentioning

confidence: 99%

“…For PW basis functions, the first proposal in [19, §2.4] for the UWVF was a local preconditioner, equivalent to an orthonormalisation of the PW basis with respect to an L 2 inner product on the boundary of mesh cells. An interesting connection of the local preconditioner with non-overlapping optimised Schwarz domain decomposition methods was discovered in [13]. The local preconditioner was used in conjunction with a BiCGStab Krylov subspace solver in [61] and augmented by a coarse-grid correction in the spirit of non-overlapping domain decomposition in [59,118].…”

Section: Ill-conditioning and Solversmentioning

confidence: 99%

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A Survey of Trefftz Methods for the Helmholtz Equation

Hiptmair

Moiola

Perugia

2016

Lecture Notes in Computational Science and Engineering

125

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Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces.We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties.One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.

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Section: Non-trefftz Methods With Oscillatory Basis Functionsmentioning

confidence: 99%

Section: Ill-conditioning and Solversmentioning

confidence: 99%

A Survey of Trefftz Methods for the Helmholtz Equation

Hiptmair

Moiola

Perugia

2016

Lecture Notes in Computational Science and Engineering

125

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Schwarz Domain Decomposition Preconditioners for Plane Wave Discontinuous Galerkin Methods

Antonietti

Perugia²,

Davide

2014

Lecture Notes in Computational Science and Engineering

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This series contains monographs of lecture notes type, lecture course material, and high-quality proceedings on topics described by the term "computational science and engineering". This includes theoretical aspects of scientific computing such as mathematical modeling, optimization methods, discretization techniques, multiscale approaches, fast solution algorithms, parallelization, and visualization methods as well as the application of these approaches throughout the disciplines of biology, chemistry, physics, engineering, earth sciences, and economics.

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Analysis of the Parallel Schwarz Method for Growing Chains of Fixed-sized Subdomains: Part II

Ciaramella¹,

Gander²

2018

SIAM J. Numer. Anal.

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Block Jacobi Relaxation for Plane Wave Discontinuous Galerkin Methods

Cited by 3 publications

References 18 publications

A Survey of Trefftz Methods for the Helmholtz Equation

A Survey of Trefftz Methods for the Helmholtz Equation

Schwarz Domain Decomposition Preconditioners for Plane Wave Discontinuous Galerkin Methods

Analysis of the Parallel Schwarz Method for Growing Chains of Fixed-sized Subdomains: Part II

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