Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

Bertrand, Fleurianne; Demkowicz, Leszek; Gopalakrishnan, Jay

doi:10.1016/j.camwa.2021.05.029

Computers & Mathematics with Applications

2021

DOI: 10.1016/j.camwa.2021.05.029

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Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

Fleurianne Bertrand¹,

Leszek Demkowicz²,

Jay Gopalakrishnan³

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2024

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Cited by 2 publications

References 19 publications

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Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

Lu,

Jiang,

Shu

et al. 2024

Numerical Methods Partial

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In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.

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Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

Lu,

Jiang,

Shu

et al. 2024

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

Generating probability distributions on intervals and spheres: Convex decomposition

Sun,

Sun

2024

Computers & Mathematics with Applications

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Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

Cited by 2 publications

References 19 publications

Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

Generating probability distributions on intervals and spheres: Convex decomposition

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