Finite element theory on curved domains with applications to DGFEMs
For researchers in finite element methods, this work extends fundamental theoretical tools from polytopal to curved domains, enabling rigorous analysis of DGFEMs on complex geometries.
This paper provides key theoretical estimates (trace, inverse, Poincaré-Friedrichs, interpolation) for DGFEMs on curved domains, including novel curvature bounds for curved simplices, and proves optimal error estimates for the IPDG method for Poisson and an hp-DGFEM for biharmonic problems. Numerical results validate the estimates.
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 analysing the IPDG method for the Poisson problem, introduced by Douglas and Dupont [\emph{Computing Methods in Applied Sciences, Lecture Notes in Physics, vol 58. Springer, Berlin, Heidelberg}, pages 207--216. Springer, 1976], and by analysing a variant of the $hp$-DGFEM for the biharmonic problem introduced by Mozolevski and Süli [\emph{Computer Methods in Applied Mechanics and Engineering}, 196(13-16):1851--1863, 2007]. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.