Søren Jensen

On the L 2 error for the p -version of the finite element method over polygonal domains

For second order linear elliptic problems over smooth domains, it is well known that the rate of convergence of the error in the L2 norm is one order higher than that in the H1 norm. For polygonal domains with re-entrant corners, it has been shown by Wahlbin [Math. Comp. 42 (1984) 1–8] that this extra order cannot be fully recovered when the h-version is used. We prove that this deterioration does not occur for the p-version and that the full extra order is observed. Negative norm estimates are also established.

Preconditioning of the p-version of the finite element method

The p-version finite element method for linear, second-order elliptic equations in an arbitrary, sufficiently smooth (incl. polygonal), bounded domain is studied in the framework of the Domain Decomposition (DD) method. Two types of square reference elements are used with coordinate functions given by the products of the integrated Legendre polynomials. Estimates for the condition numbers and some useful inequalities are given. We consider preconditioning of the problems arising on subdomains and of the Schur complement, as well as the derivation and analysis of the DD preconditioner for the entire system. This is done for a class of curvilinear finite elements. We obtain several DD preconditioners for which the generalized condition numbers vary from O((log p)3) to O(1). This paper is based on [19–21,27]. We have omitted most of the proofs in order to shorten it and have described instead what could be done as well as outlined some additional ideas. The full proofs omitted can in most cases be found in [19,20,27].

The Kaccanov method for some nonlinear problems

Abstract The Kacanov method is an iteration method for solving some nonlinear partial differential equation problems. In each iteration, a linear problem is solved. In this paper, we discuss the use of the Kacanov method in the context of two model problems. We show the convergence of the Kacanov iteration sequences, and derive a posteriori error estimates for the Kacanov iterates. Numerical examples are given showing the convergence of the method and the effectiveness of the a posteriori error estimates.

p -version of mixed finite element methods for Stokes-like problems

Abstract We investigate various sample extremum (saddle point and minimization) problems from the point of view of stability of increasing order mixed finite element methods. Problems discussed include the Stokes, Poisson and linear elasticity problems.

Behaviour in the large on numerical solutions to one-dimensional nonlinear viscoelasticity by continuous time Galerkin methods

Abstract We analyze the long time behaviour of fully discrete solutions to a one-dimensional nonlinear viscoelastic problem. It is shown that these approximations which are found by a continuous time Galerkin method converge to a steady state. The possible numerical steady states are characterized and in particular their high degree of dependence on initial data and mesh design is explained. Computational results are included which show the above dependence and indicate that the numerical solutions will typically not converge to unstable states.

Finite element approximation of solutions to a class of nonlinear hyperbolic-parabolic equations

Abstract The numerical approximation of Antman and Seidmans (1996) model of the longitudinal motion of a viscoelastic rod is investigated. Their constitutive assumptions ensure that infinite compressive stress is needed to produce total compression of the rod. Analyses of the regularity of the solution of the continuous problem, the convergence of a semi-discrete finite element method, and the properties of a space–time finite element scheme are furnished. Results of a sample computation are also provided.

On the stability and performance of the p -version of the finite element method for first-order hyperbolic problems

We study the p-version of the finite element method for the first-order scalar hyperbolic equation with constant coefficients. We analyze the stability and convergence of the method. A comparison with the discontinuous h-version and results of numerical experiments are also presented.