Abstract. 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 displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.
We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s − r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r − 2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.
Abstract. We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the serendipity spaces, and new H(curl) and H(div) finite element spaces. Spaces in the family can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, and hence can be used for stable discretization of a variety of problems. The construction and properties of the spaces are established in a uniform manner using finite element exterior calculus.
Abstract. We present numerical approximations of the 3D steady state Navier-Stokes equations in velocity-pressure formulation using trivariate splines of arbitrary degree d and arbitrary smoothness r with r < d. Using functional arguments, we derive the discrete Navier-Stokes equations in terms of Bcoefficients of trivariate splines over a tetrahedral partition of any given polygonal domain. Smoothness conditions, boundary conditions and the divergencefree condition are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and present numerical evidence of the convergence rate as well as experiments on the lid driven cavity flow problem.
We present a family of stable rectangular mixed finite elements for plane elasticity. Each member of the family consists of a space of piecewise polynomials discretizing the space of symmetric tensor fields in which the stress field is sought, and another to discretize the space of vector fields in which the displacement is sought. These may be viewed as analogues in the case of rectangular meshes of mixed finite elements recently proposed for triangular meshes. As for the triangular case the elements are closely related to a discrete version of the elasticity differential complex.
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