Selection Strategies in Adaptive Refinement

Abstract

This paper deals with the problem of selecting the elements to be refined for the construction of a new triangulation in an adaptive refinement system. The problem considered is the
numerical solution of Poisson's equation using piecewise linear finite elements and local error indicators of Babuska-Miller-type. We analyze two ways of selecting triangles in adaptive refinement: the first strategy (widely used) is to mark elements that have an indicator greater than α times the largest of the indicators where 0≤α ≤1. We conclude that this method is robust in the following sense: if we choose α≥αo with αo small, the convergence order of the regular problem with quasi uniform meshes (measured with the number of elements) is recovered. In this procedure we also introduce a stopping criterion to obtain the final error measure smaller than a prescribed tolerance. The second strategy is to mark elements that have an indicator greater than an admissible indicator. This admissible indicator is defined based on the previous stopping criterion. The ratio between the elemental indicator and the admissible error is also used to define the level of refinement in each element. We analyze the behavior of both strategies and compare them. Finally, some remarks about the whole adaptive process are discussed.