Quasi-optimal Nonconforming Approximation of Elliptic PDEs with Contrasted Coefficients and $$H^{1+{r}}$$, $${r}>0$$, Regularity
| Autor: | Alexandre Ern, Jean-Luc Guermond |
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| Přispěvatelé: | Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Department of Mathematics [Texas] (TAMU), Texas A&M University [College Station], This material is based upon work supported in part by the National Science Foundation grants DMS-1619892, DMS-1620058, by the Air Force Office of Scientific Research, USAF, undergrant/contract number FA9550-18-1-0397, and by the Army Research Office under grant/contract number W911NF-15-1-0517. |
| Rok vydání: | 2021 |
| Předmět: |
Nitsche's method
Nonconforming methods Smoothness (probability theory) Applied Mathematics Numerical analysis Finite elements Elliptic equations Directional derivative Sobolev space Computational Mathematics Maxwell's equations AMS subject classifications.35J25 65N15 65N30 Computational Theory and Mathematics Minimal regularity Approximation error Bounded function Norm (mathematics) Applied mathematics Error estimates Constant (mathematics) [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] Analysis discontinuous Galerkin Mathematics |
| Zdroj: | Foundations of Computational Mathematics Foundations of Computational Mathematics, 2021, 22 (5), pp.1273--1308. ⟨10.1007/s10208-021-09527-7⟩ |
| ISSN: | 1615-3383 1615-3375 |
| DOI: | 10.1007/s10208-021-09527-7 |
| Popis: | International audience; In this paper, we investigate the approximation of a diffusion modelproblem with contrasted diffusivity for various nonconformingapproximation methods. The essential difficulty is that the Sobolevsmoothness index of the exact solution may be just barely largerthan 1. The lack of smoothness is handled by giving a weak meaningto the normal derivative of the exact solution at the mesh faces.We derive robust and quasi-optimal error estimates.Quasi-optimality means that the approximation error is bounded,up to a generic constant, by the best-approximation error in thediscrete trial space, and robustness means that the genericconstant is independent of the diffusivity contrast. The errorestimates use a mesh-dependent norm that is equivalent, at thediscrete level, to the energy norm and that remains bounded aslong as the exact solution has a Sobolev index strictly largerthan 1. Finally, we briefly show how the analysis can be extendedto the Maxwell's equations. |
| Databáze: | OpenAIRE |
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