A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex Academic Article uri icon

abstract

  • 2014, SFoCM. 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 includes a well-developed theory of finite element methods for HodgeLaplace problems, including apriori error estimates. In this work we focus on developing aposteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove aposteriori error estimates of a residual type for ArnoldFalkWinther mixed finite element methods for Hodgede RhamLaplace problems. While a number of previous works consider aposteriori error estimation for Maxwells equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various HodgeLaplace problems arising in the deRham complex, consistent use of the language and analytical framework of differential forms, and the development of aposteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the HodgeLaplacian.

published proceedings

  • Foundations of Computational Mathematics

author list (cited authors)

  • Demlow, A., & Hirani, A. N.

citation count

  • 24

complete list of authors

  • Demlow, Alan||Hirani, Anil N

publication date

  • December 2014