Gerard Awanou

FINITE ELEMENTS FOR SYMMETRIC TENSORS IN THREE DIMENSIONS

We construct finite element subspaces of the space of symme- tric 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 stan- dard discontinous finite element spaces are used to approximate the displace- ment 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.

The Serendipity Family of Finite Elements

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.

RECTANGULAR MIXED FINITE ELEMENTS FOR ELASTICITY

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.

Finite element differential forms on cubical meshes

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.

Trivariate spline approximations of 3D Navier-Stokes equations

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 B-coefficients of trivariate splines over a tetrahedral partition of any given polygonal domain. Smoothness conditions, boundary conditions and the divergence-free 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.

Pseudo transient continuation and time marching methods for Monge-Ampère type equations

We present two numerical methods for the fully nonlinear elliptic Monge-Ampère equation. The first is a pseudo transient continuation method and the second is a pure pseudo time marching method. The methods are proven to converge to a strictly convex solution of a natural discrete variational formulation with C1 conforming approximations. The assumption of existence of a strictly convex solution to the discrete problem is proven for smooth solutions of the continuous problem and supported by numerical evidence for non smooth solutions.

Two Remarks on Rectangular Mixed Finite Elements for Elasticity

The lowest order nonconforming rectangular element in three dimensions involves 54 degrees of freedom for the stress and 12 degrees of freedom for the displacement. With a modest increase in the number of degrees of freedom (24 for the stress), we obtain a conforming rectangular element for linear elasticity in three dimensions. Moreover, unlike the conforming plane rectangular or simplicial elements, this element does not involve any vertex degrees of freedom. Second, we remark that further low order elements can be constructed by approximating the displacement with rigid body motions. This results in a pair of conforming elements with 72 degrees of freedom for the stress and 6 degrees of freedom for the displacement.

Rectangular mixed elements for elasticity with weakly imposed symmetry condition

We present new rectangular mixed finite elements for linear elasticity. The approach is based on a modification of the Hellinger–Reissner functional in which the symmetry of the stress field is enforced weakly through the introduction of a Lagrange multiplier. The elements are analogues of the lowest order elements described in Arnold et al. (Math Comput 76:1699–1723, 2007). Piecewise constants are used to approximate the displacement and the rotation. The first order BDM elements are used to approximate each row of the stress field.

On Standard Finite Difference Discretizations of the Elliptic Monge---Ampère Equation

Given an orthogonal lattice with mesh length h on a bounded two-dimensional convex domain