An optimal control approach to a posteriori error estimation in finite element methods

Abstract

This article surveys a general approach to error control and adaptive mesh design in Galerkin finite element methods that is based on duality principles as used in optimal control. Most of the existing work on a posteriori error analysis deals with error estimation in global norms like the ‘energy norm’ or the L2 norm, involving usually unknown ‘stability constants’. However, in most applications, the error in a global norm does not provide useful bounds for the errors in the quantities of real physical interest. Further, their sensitivity to local error sources is not properly represented by global stability constants. These deficiencies are overcome by employing duality techniques, as is common in a priori error analysis of finite element methods, and replacing the global stability constants by computationally obtained local sensitivity factors. Combining this with Galerkin orthogonality, a posteriori estimates can be derived directly for the error in the target quantity. In these estimates local residuals of the computed solution are multiplied by weights which measure the dependence of the error on the local residuals. Those, in turn, can be controlled by locally refining or coarsening the computational mesh. The weights are obtained by approximately solving a linear adjoint problem. The resulting a posteriori error estimates provide the basis of a feedback process for successively constructing economical meshes and corresponding error bounds tailored to the particular goal of the computation. This approach, called the ‘dual-weighted-residual method’, is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems. After having discussed the basic properties of duality-based adaptivity, we demonstrate the potential of this approach by presenting a selection of results obtained for practical test cases. These include problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative transfer, and optimal control. Throughout the paper, open theoretical and practical problems are stated together with references to the relevant literature.

A review of numerical methods for nonlinear partial differential equations

“…Stability of mixed-type FEMs for saddle-point problems is verified in terms of the Babuška-Brezzi inf-sup condition [8,26]. The finite elements are adapted to local variations of the solution: adaptivity is often tuned by a posteriori estimates [14,22,157].…”

Adaptive Finite Element Approximation of Fluid-Structure Interaction Based on an Eulerian Variational Formulation

“…This method is explained in [4] (see also [1]) as an extension of the duality technique for a posteriori error estimation described in [12]. The DWR method provides a general framework for the derivation of 'goal-oriented' a posteriori error estimates together with criteria of mesh adaptation for the Galerkin discretization of general linear and nonlinear variational problems, including optimization problems.…”

“…Based on the Eulerian variational formulation of the FSI system, we use the 'dual weighted residual' (DWR) method, described in [4,1], to derive 'goal-oriented' a posteriori error estimates. The evaluation of these error estimates requires the approximate solution of a linear dual variational problem.…”

A Computational Infrastructure for Reliable Computer Simulations

“…A general theory for controling this type of modeling error in quantities of interest Q through a posteriori error estimation and adaptive modeling has been developed by Oden and Prudhomme [19,17], and Oden and Vemaganti [15,16]. Techniques for deriving a posteriori error estimates for ε approx and goal-oriented adaptive meshing have been advanced by Babuska and Strouboulis [3], Oden and Prudhomme [18,19], Rannacher, Becker and others [9]. Thus we assume that using the data available, it is possible to obtain computable lower and upper bounds on these components:…”

“…Verification is thus concerned with estimating and controling numerical approximation error. In recent years, significant progress has been made in this area (see Ainsworth and Oden [1], Babuska and Strouboulis [3], Oden and Prudhomme [17], Becker and Rannacher [9]). …”