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Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods.
- Source :
- Numerical Methods for Partial Differential Equations; Nov2020, Vol. 36 Issue 6, p1492-1536, 45p
- Publication Year :
- 2020
-
Abstract
- In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp‐DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates. [ABSTRACT FROM AUTHOR]
- Subjects :
- FINITE element method
ERROR analysis in mathematics
PARTIAL differential equations
Subjects
Details
- Language :
- English
- ISSN :
- 0749159X
- Volume :
- 36
- Issue :
- 6
- Database :
- Complementary Index
- Journal :
- Numerical Methods for Partial Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 146104538
- Full Text :
- https://doi.org/10.1002/num.22489