Adaptive finite element methods for partial differential equations

TL;DR

This paper discusses an adaptive finite element method using dual weighted residuals to efficiently estimate errors and optimize meshes for nonlinear PDEs, demonstrated through fluid mechanics applications.

Contribution

It introduces a dual weighted residual approach for a posteriori error estimation in finite element methods, enabling goal-oriented mesh adaptation for nonlinear problems.

Findings

Effective error estimation for nonlinear PDEs

Adaptive meshes improve computational efficiency

Successful application to fluid mechanics problems

Abstract

The numerical simulation of complex physical processes requires the use of economical discrete models. This lecture presents a general paradigm of deriving a posteriori error estimates for the Galerkin finite element approximation of nonlinear problems. Employing duality techniques as used in optimal control theory the error in the target quantities is estimated in terms of weighted `primal' and `dual' residuals. On the basis of the resulting local error indicators economical meshes can be constructed which are tailored to the particular goal of the computation. The performance of this {\it Dual Weighted Residual Method} is illustrated for a model situation in computational fluid mechanics: the computation of the drag of a body in a viscous flow, the drag minimization by boundary control and the investigation of the optimal solution's stability.