Richard S. Falk

TL;DR: Finite element exterior calculus as mentioned in this paper is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations, which brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological and algebraic structures which underlie well-posedness of the PDE problem being solved.

TL;DR: In this article, the authors consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem.

TL;DR: In this article, the authors developed an abstract Hilbert space framework for analyzing stability and convergence of finite element approximations of the Hodge Laplacian in the continuous problem.

TL;DR: If appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator.

TL;DR: In this article, a general approximation theorem useful in obtaining order of convergence estimates for the approximation of the solutions of a class of variational inequalities is presented. And the theorem is then applied to obtain an "optimal" rate of convergence for approximation of a second-order elliptic problem with convex set K = {v E Ho(): v > X a.