Document Zbl 1376.65145

Müller, Fabian; Schötzau, Dominik; Schwab, Christoph

Symmetric interior penalty discontinuous Galerkin methods for elliptic problems in polygons. (English) Zbl 1376.65145

SIAM J. Numer. Anal. 55, No. 5, 2490-2521 (2017).

Summary: We analyze symmetric interior penalty discontinuous Galerkin (DG) finite element methods for linear, second-order elliptic boundary-value problems in polygons \(\Omega\) with straight edges where solutions exhibit singular behavior near corners, and at boundary points where boundary conditions change. To resolve corner singularities, we admit both graded meshes and bisection refinement meshes. We prove that judiciously chosen refinement parameters in these mesh families imply optimal asymptotic rates of convergence with respect to the total number of degrees of freedom \(N\), both for the DG energy norm error and the \(L^2\)-norm error. The sharpness of our asymptotic convergence rate estimates is confirmed in a series of numerical experiments.

Cited in 5 Documents

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N15	Error bounds for boundary value problems involving PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs

Keywords:

elliptic boundary-value problems; finite element methods; discontinuous Galerkin methods; corner singularities; Lipschitz domains; error bounds; convergence; numerical experiment

Software:

SciPy; Triangle; LNG_FEM

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Full Text: DOI

References:

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