Quasi-Optimal Mesh Sequence Construction through Smoothed Adaptive Finite Element Methods

Adaptive Finite Element Methods for Optimal

Adaptive Finite Element Methods for Optimal Control

We develop a new approach towards error control and adaptivity for nite element discretizations in optimization problems governed by partial diierential equations. Using the Lagrangian formalism the goal is to compute stationary points of the rst-order necessary opti-mality conditions. The mesh adaptation is driven by residual-based a posteriori error estimates derived by duality arguments. Thi...

Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method

We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of th...

Adaptive Finite Element Methods

In the numerical solution of practical problems of physics or engineering such as, e.g., computational fluid dynamics, elasticity, or semiconductor device simulation one often encounters the difficulty that the overall accuracy of the numerical approximation is deteriorated by local singularities such as, e.g., singularities arising from re-entrant corners , interior or boundary layers, or shar...

Adaptive Finite Element Methods

Adaptive methods are now widely used in the scientific computation to achieve better accuracy with minimum degree of freedom. In this chapter, we shall briefly survey recent progress on the convergence analysis of adaptive finite element methods (AFEMs) for second order elliptic partial differential equations and refer to Nochetto, Siebert and Veeser [14] for a detailed introduction to the theo...