Susanne C. Brenner

Poincaré-Friedrichs Inequalities for Piecewise H 1 Functions

Poincare--Friedrichs inequalities for piecewise H1 functions are established. They can be applied to classical nonconforming finite element methods, mortar methods, and discontinuous Galerkin methods.

Korn's inequalities for piecewise

Korns inequalities for piecewise H 1 vector fields are established. They can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.

H^1

C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are analyzed in this paper. A post-processing procedure that can generate C1 approximate solutions from the C0 approximate solutions is presented. New C0 interior penalty methods based on the techniques involved in the post-processing procedure are introduced. These new methods are applicable to rough right-hand sides.

vector fields

A linear nonconforming (conforming) displacement finite element method for the pure displacement (pure traction) problem in two-dimensional linear elasticity for a homogeneous isotropic elastic material is considered. In the case of a convex polygonal configuration domain, error estimates in the energy (L[sup 2]) norm are obtained. The convergence rate does not deteriorate for nearly incompressible material. Furthermore, the convergence analysis does not rely on the theory of saddle point problems. 22 refs.

C0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains

Two-level additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar second-order symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergence-free nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap.

Linear finite element methods for planar linear elasticity

We consider the Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks. Multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed. The convergence rate for the stress intensity factors is\(\mathcal{O}(h^{(3/2) - \in } )\) whenfeL2(Ω) and\(\mathcal{O}(h^{(2 - \in )} )\) whenfeH1(Ω). The convergence rate in the energy norm is\(\mathcal{O}(h^{(1 - \in )} )\) in the first case and\(\mathcal{O}(h)\) in the second case. The costs of these multigrid methods are proportional to the number of elements in the triangulation. The general case wherefeHm(Ω) is also discussed.

Two-level additive Schwarz preconditioners for nonconforming finite element methods

We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the W-cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable V-cycle algorithm is an optimal preconditioner.

Multigrid methods for the computation of singular solutions and stress intensity factors I: corner singularities

An optimal order multigrid method for the lowest-order Raviart–Thomas mixed triangular finite element is developed. The algorithm and the convergence analysis are based on the equivalence between Raviart–Thomas mixed methods and certain nonconforming methods. Both the Dirichlet and singular Neumann boundary value problems for second-order elliptic equations are discussed.

Convergence of nonconforming multigrid methods without full elliptic regularity

In this paper, we develop and analyze C(0) penalty methods for the fully nonlinear Monge-Ampere equation det(D(2)u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.

A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method

More than ten years ago, the finite element method, a modern, systematic numerical method for solving differential equations, was created and developed independently and along different lines in China[6] and the West. Its original purpose was to solve for equilibria and stable configurations, i.e. to solve elliptic equations. It has stood a large number of practical tests, and, in particular, has been used widely in the field of elastic structures with remarkable success. Recently, with the aid of computers, the finite element method has been applied to almost all fileds of engineering and to many fields of science and technology, and has become a routine means for modern engineering analysis. It is an important achievement of modern computational mathematics.