A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

作者：Demlow Alan^*; Hirani Anil N

来源：Foundations of Computational Mathematics, 2014, 14(6): 1337-1371.

DOI：10.1007/s10208-014-9203-2

• Finite element methods
• Exterior calculus
• A posteriori error estimates
• Adaptivity

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 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 a 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%26apos;s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a 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.

• 出版日期2014-12