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 sharp shock-like fronts. An obvious remedy is to refine the discretization near the critical regions, i.e., to place more grid-points where the solution is less regular. The question then is how to identify those regions and how to obtain a good balance between the refined and unrefined regions such that the overall accuracy is optimal. Another closely related problem is to obtain reliable estimates of the accuracy of the computed numerical solution. A priori error estimates, as provided, e.g., by the standard error analysis for finite element or finite difference methods, are often insufficient since they only yield information on the asymptotic error behaviour and require regularity conditions of the solution which are not satisfied in the presence of singularities as described above. These considerations clearly show the need for an error estimator which can a posteriori be extracted from the computed numerical solution and the given data of the problem. Of course, the calculation of the a posteriori error estimate should be far less expensive than the computation of the numerical solution. Moreover, the error estimator should be local and should yield reliable upper and lower bounds for the true error in a user-specified norm. In this context one should note, that global upper bounds are sufficient to obtain a numerical solution with an accuracy below a prescribed tolerance. Local lower bounds, however, are necessary to ensure that the grid is correctly refined so that one obtains a numerical solution with a prescribed tolerance using a (nearly) minimal number of grid-points. Disposing of an a posteriori error estimator, an adaptive mesh-refinement process has the following general structure: Algorithm I.1.1. (General adaptive algorithm) (0) Given: The data of a partial differential equation and a tolerance ε.