Accounting for stability: a posteriori error estimates based on residuals and variational analysis

Abstract

Abstract. A posteriori error estimates have had a major impact on adaptive error control for the finite element method. In this paper, we review a relatively new approach to a posteriori error estimation based on residuals and a variational analysis. This approach is distinguished by a direct attempt to account for the effects of stability on the propagation of error. We illustrate properties of this approach using several examples.Key words. a posteriori error estimate, adaptive error control, computational error estimation, dual problem, finite element method, multi-scaled problems, residual, stability, stability factor, variational analysis 1. Introduction. The search for reliably accurate numerical tools for multi-scaled differential equations has become increasingly urgent in recent years. Not the least because multi-scaled problems arise in a wide range of applications and computing accurate numerical solutions often strains the limits of computational resources. One way to obtain accurate solutions of multi-scaled problems is through computational error estimation and adaptive error control. Computational error estimation is directed towards determining the kind of information that can be accurately obtained from a particular computation and estimating the accuracy of said information. This is an important goal both in general scientific terms as well as in terms of adaptive error control, that is for deciding how to use computational resources to achieve a desired accuracy.Over the last two decades, there has been significant progress in computational error estimation and adaptive error control arising out of developments in a posteriori error analysis. In a posteriori analysis, the error of a numerical solution is estimated as much as possible in terms of computable quantities that depend on the numerical solution and in particular the estimate is computed after the numerical solution has been computed. The progress in a posteriori analysis has resulted in important advances in the reliably accurate solution of multi-scaled problems. As a consequence, a posteriori techniques have found widespread use in engineering and mathematics.There are several different approaches to a posteriori error analysis. In this paper, we review a relatively new approach based on residuals and variational analysis involving the dual, or adjoint, problem to the original equation. This approach is distinguished by a direct attempt to account for the effects of propagation and accumulation of errors by computational means. We present a formal description of this approach and describe the application to elliptic and parabolic differential equations. We also illustrate various aspects of this approach using a set of numerical examples.

Duality-based two-level error estimation for time-dependent PDEs: Application to linear and nonlinear parabolic equations

“…Moreover, for nonlinear problems, it can be very hard to obtain quantitatively-accurate estimates because the invoked bounds typically consider worst-case scenarios, leading to huge stability constants. In this regard, we agree with Estep, Holst, and Mikulencak [16]: "[Classical estimation] is generally frustrating, [. .…”

Convergence of goal‐oriented adaptive finite element methods for nonsymmetric problems

“…We also show a number of the adaptive meshes below, demonstrating that in many cases, the compared methods produce similar performance from qualitatively different adaptive refinements. A discussion of the DWR method may be found in [2,4,9,13,14,10] for example. In our DWR implementation, the finite element space for the primal problem V T k employs linear Lagrange elements as do HP and MS for both the primal and dual spaces.…”

“…Our analysis approach is signficantly different from that of Mommer and Stevenson [18], combining the recent contraction frameworks of Cascon, Kreuzer, Nochetto and Siebert [7], of Nochetto, Siebert and Veeser [19], and of Holst, Tsogtgerel and Zhu [16]. We also give some numerical results comparing our goal-oriented method both to the one presented in [18] and the dual weighted residual (DWR) method as in [2,4,9,13,14,10], among others. Unlike the existing literature on the DWR method, we prove strong convergence of our goal-oriented method.…”

Reordering and incomplete preconditioning in serial and parallel adaptive mesh refinement and coarsening flow solutions

“…The behavior of AMR/C with incomplete factorization preconditioners under node reorderings for serial and then parallel implementations are of particular interest. Other more sophisticated error indicators based on patch recovery [26,31] or adjoint methods [32][33][34] can be utilized, but the choice of optimal indicator is not in the scope of this contribution. The indicator is based on the jump in the normal gradient across the interface between adjacent elements and is similar to the indicator proposed in [25].…”