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We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the dis-continuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems. 1. Introduction. In 1973, Reed and Hill [61]… (More)

- Douglas N. Arnold, Richard S Falk, Ragnar Winther, D N Arnold, R S Falk, R Winther
- 2006

Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza-tions which are compatible with the geometric,… (More)

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation , the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are… (More)

There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the… (More)

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger–Reissner variational principle that only weakly imposes the symmetry condition on the… (More)

We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces H(div) and H(curl) in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is… (More)

- Douglas N. Arnold, Richard S Falk, And R Winther
- 1997

We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I − grad div. The natural setting for such problems is in the Hilbert space H (div) and the variational formulation is based on the inner product in H (div). We show how… (More)

We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger–Reissner mixed formulation of the elasticty equations, when standard discontinuous finite element spaces are used to approximate the… (More)

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element , which is then transformed to a space of functions on each convex quadri-lateral element via a bilinear isomorphism of the square onto… (More)

We consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, extending results previously obtained for scalar approximation. The finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element, which is then transformed to… (More)