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- 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)

- Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, Donatella Marini
- 2000

We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the variants proposed by Brezzi, Manzini, Marini, Pietra and Russo), the local… (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)

Summarizing the work of [AFW97], we show how to construct preconditioners using domain decomposition and multigrid techniques for 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. These preconditioners are shown to be spectrally… (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)

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)

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)