A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex
| Autor: | Alan Demlow, Anil N. Hirani |
|---|---|
| Rok vydání: | 2014 |
| Předmět: |
Partial differential equation
Differential form Applied Mathematics Numerical analysis Mathematical analysis Numerical Analysis (math.NA) 010103 numerical & computational mathematics Mixed finite element method Residual 01 natural sciences Finite element method Mathematics::Numerical Analysis 010101 applied mathematics Computational Mathematics Finite element exterior calculus Computational Theory and Mathematics 65N15 65N30 FOS: Mathematics A priori and a posteriori Mathematics - Numerical Analysis 0101 mathematics Analysis Mathematics |
| Zdroj: | Foundations of Computational Mathematics. 14:1337-1371 |
| ISSN: | 1615-3383 1615-3375 |
| Popis: | Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther \cite{ArFaWi2010} includes a well-developed theory of finite element methods for Hodge Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of residual type for Arnold-Falk-Winther mixed finite element methods for Hodge-de Rham Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell's equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by unified treatment of the various Hodge Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge Laplacian. Comment: 30 pages |
| Databáze: | OpenAIRE |
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