Adaptive Finite Element Methods for Parabolic Problems II: Optimal Error Estimates in $L_\infty L_2 $ and $L_\infty L_\infty $

Abstract

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.Key words, parabolic problem, adaptive finite element method, discontinuous Galerkin method, a priori and a posteriori error estimates GSteborg, 5-412 96 GSteborg, Sweden (Kenneth(C)math.chalmers. 706 Downloaded 11/26/14 to 18.101.24.154. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ADAPTIVE FEM FOR PARABOLIC PROBLEMS II 707Galerkin procedures for such problems. We prove this result to be true not only for strictly coercive problems where all eigenvalues of the corresponding linearized elliptic operator are strictly positive but also in cases of slightly negative eigenvalues, which opens possibilities of application, e.g., to reaction-diffusion problems (see [8]). The basic tool to obtain these results is a new stability concept together with the Galerkin orthogonalities built into the finite element method, as discussed further below. See also [10].Our adaptive methods are based on a posteriori error estimates where the error is estimated in terms of a directly computable quantity depending on the local mesh size in time and space, the given data, and the computed solution. The adaptive

A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method

“…The relation (1.4) is an optimal-order error estimate provided we do not change the spaces S n h too often-compare with the results of Dupont [D]; but even in this case (1. [EJ2], show that optimal order-regularity estimates (up to a logarithmic factor) are valid for the corresponding discontinuous Galerkin method for linear parabolic problems, provided in each time slab either S n h ⊂ S n−1 h or k n ≥ ch 2 n . The approach in [EJ1], [EJ2] is based on establishing "strong" stability estimates for an appropriate discrete dual problem.…”

“…[EJ2], show that optimal order-regularity estimates (up to a logarithmic factor) are valid for the corresponding discontinuous Galerkin method for linear parabolic problems, provided in each time slab either S n h ⊂ S n−1 h or k n ≥ ch 2 n . The approach in [EJ1], [EJ2] is based on establishing "strong" stability estimates for an appropriate discrete dual problem. These estimates depend in an essential manner on the parabolic character of the problem considered in [EJ1], [EJ2].…”

“…In higher dimensions (3.4b) is valid under appropriate local quasiuniformity conditions, cf. [EJ1], [EJ2] and their references and also [BS, Chapter 0] for a discussion of the one-dimensional case. Note that (3.4b) is not an essential assumption for our results to hold, in the sense that our estimates will remain valid if we replace the h r n v r,h -like terms in Theorems 3.1 and 4.1 by terms that bound the error of the elliptic projection in the L 2 -norm.…”

A priori error analysis for finite element approximations of the Stokes problem on dynamic meshes

“…Let us mention the work of Dupont (1982), Eriksson & Johnson (1991, 1995 and Yang (2000). In the pioneering work of Dupont (1982), for a one-dimensional parabolic model problem, it is shown that changing the mesh in a particular way in each time step leads to convergence of the numerical solution to a wrong answer.…”

“…In the pioneering work of Dupont (1982), for a one-dimensional parabolic model problem, it is shown that changing the mesh in a particular way in each time step leads to convergence of the numerical solution to a wrong answer. In Eriksson & Johnson (1991, 1995 an a priori error analysis based on duality arguments is given for finite element approximations of a class of parabolic problems. Optimal-order error estimates are established under restrictions on the mesh changes, which have to satisfy either that the finite element spaces are embedded, T n ⊂ T n−1 , or that the restriction h 2 n CΔt n on the discretization step sizes is satisfied, where h n and Δt n denote the spatial and temporal step size, respectively, and C is a constant.…”