Kenneth Eriksson

Adaptive finite element methods for parabolic problems. I.: a linear model problem

This paper is the first part in a series of papers on adaptive finite element methods for parabolic problems. In this paper, an adaptive algorithm is presented and analyzed for choosing the space a...

Introduction to Adaptive Methods for Differential Equations

Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646–1719). When, severa ...

Adaptive finite element methods for parabolic problems IV: nonlinear problems

We extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems. We prove a posteriori error estimates, design corresponding adaptive algorithms, and present some numerical results.

Adaptive finite element methods for parabolic problems II: optimal error estimates in L ∞ L 2 and L ∞ L ∞

Optimal error estimates are derived for a complete discretization of linear parabolic problems using space–time finite elements. The discretization is done first in time using the discontinuous Galerkin method and then in space using the standard Galerkin method. The underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step. The error bounds show, in particular, that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same (given) level, i.e., without error accumulation effects. The derivation of the estimates is based on the orthogonality of the Galerkin procedure and the use of strong stability estimates. The particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration.

Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems

Adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed. The underlying discretization scheme is the Shock-capturing Streamline Diffusion method. The adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed. A priori error estimates are used to show that the algorithms are efficient in a certain sense.

Adaptive finite element methods for parabolic problems V: long-time integration

We continue our previous work on adaptive finite element methods for parabolic problems, now with particular emphasis on long-time integration for semidefinite problems.

Adaptive Finite Element Methods for Parabolic Problems VI: Analytic Semigroups

We continue our work on adaptive finite element methods with a study of time discretization of analytic semigroups. We prove optimal a priori and a posteriori error estimates for the discontinuous Galerkin method showing, in particular, that analytic semigroups allow long-time integration without error accumulation.

Explicit Time-Stepping for Stiff ODEs

We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis. For many stiff problems the cost of the stabilizing small time steps is small, so the improvement is large. We illustrate the technique on a number of well-known stiff test problems.

On adaptive finite element methods for Fredholm integral equations of the second kind

A posteriors and a priori error estimates are derived for a finite element discretization of a Fredholm integral equation of the second kind. A reliable and efficient adaptive algorithm is then designed for a specific computational goal with applications to potential problems. The reliability of the algorithm is guaranteed by the a posteriors error estimate and the efficiency follows from the a priori error estimate, which shows that the a posteriors error bound is sharp

Some error estimates for the p -version of the finite element method

In the p-version of the finite element method the trial and test function spaces consist of piecewise polynomial functions of sufficiently high order on a coarse partition of the domain into a few convex elements. In this note the p-method is applied to a one-dimension model problem the solution of which has a singularity similar to that of the solution of a two-dimensional Dirichlet type corner problem for an elliptic equation. Error estimates are derived in the energy norm, the