Related papers: Convergence and Optimality of Adaptive Mixed Metho…
A recent paper of Arnold, Falk, and Winther [Bull AMS, 47 (2010)] showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational…
Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can…
Finite element exterior calculus (FEEC) has been developed as a systematical framework for constructing and analyzing stable and accurate numerical method for partial differential equations by employing differential complexes. This paper is…
Over the last ten years, the Finite Element Exterior Calculus (FEEC) has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic…
Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281--354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert…
In this paper, we present several new a posteriori error estimators and two adaptive mixed finite element methods \textsf{AMFEM1} and \textsf{AMFEM2} for the Hodge Laplacian problem in finite element exterior calculus. We prove that…
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…
In this article we develop convergence theory for a class of goal-oriented adaptive finite element algorithms for second order nonsymmetric linear elliptic equations. In particular, we establish contraction results for a method of this type…
Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010), 281-354] that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract…
A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the…
The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite element-type numerical methods for the resulting coupled nonlinear…
With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent…
In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several…
Trace finite element methods have become a popular option for solving surface partial differential equations, especially in problems where surface and bulk effects are coupled. In such methods a surface mesh is formed by approximately…
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…
Proofs of convergence of adaptive finite element methods for the approximation of eigenvalues and eigenfunctions of linear elliptic problems have been given in a several recent papers. A key step in establishing such results for multiple…
Numerical computation of harmonic forms (typically called harmonic fields in three space dimensions) arises in various areas, including computer graphics and computational electromagnetics. The finite element exterior calculus framework…
We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem may be singular, which has prompted us to conduct an a posteriori analysis of the method deriving residual based estimators to drive an…
In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating the spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved…
We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…