Selection Strategies in Adaptive Refinement

Manuscript, pp.1-9

MESHING
RESEARCH
CORNER

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 alpha times the largest of the indicators where alpha is between 0 and 1. We conclude that this method is robust in the following sense: if we choose alpha greater than alpha0 with alpha0 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. In order to reduce the global computational cost we introduce an improvement of the second strategy. Finally, some remarks about the whole adaptive process are discussed.