A‐posteriori error estimates for the finite element method

Abstract

SUMMARYComputable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h + O where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are Straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.

A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems

“…Residual-based a posteriori error estimates for elliptic problems have been introduced in [4], [5] and their theory for elliptic problems is now fairly completely established (cf. [18], [1]).…”

On the a posteriori error analysis for equations of prescribed mean curvature

“…In such analysis one tries to determine computable quantities that can be used to guide an adaptive procedure. For an overview of a posteriori error analyses and their applications, we refer to the seminal papers of Babuška and Rheinboldt [BR78a,BR78b], the books of Ainsworth and Oden [AO00] and Verfürth [Ver96], and the more recent developments of Dörfler [Dör96] and Morin et al [MNS00,MNS].…”

Adjoint-Based a Posteriori Error Estimation for Coupled Time-Dependent Systems

“…We consider an adjoint-based a posteriori error estimator for parabolic problems that are coupled across a given interface and construct an adaptive space-time finite element scheme. A posteriori error estimation is commonly used in parabolic problems for adaptive mesh refinement and the different approaches are described in a detailed review paper [1], as well as in [4,5,14,20,24,35,39,40,41]. In particular, [14,15,16] develop a technique based on solving an adjoint problem to estimate the error in a quantity of interest given by a functional of the solution as opposed to the error in a norm of the solution.…”