ABSTRACT
An adaptive mesh-refining algorithm usually consists of the following loop:
Solve → Estimate → Mark → Refine.
Solve. This step computes the piecewise polynomial finite element approximation with respect to a given mesh.
Estimate. Given a partition Th and the corresponding output from the “Solve” step, “Estimate” computes some a posteriori error estimator.
Mark. We will replace the subscript h (or h k ) by an iteration counter k whenever convenient afterwards. Based on the a posteriori error indicators, “Mark” gives a strategy to choose a subset of elements ℳk of τk for refinement. One of the most widely used marking strategies to enforce error reduction is the so-called Dörfler strategy. A weaker strategy, which is called “Maximum Strategy,” only requires that the set of marked elements ℳk contains at least one element of τk holding the largest value estimator. Note that the most commonly used marking strategies, e.g., Dörfler strategy and Equidistribution strategy, fulfill this condition.
Refine. Given the partition τk and the set of marked elements ℳk , “Refine” produces a new partition τ k+1 by refining all elements in ℳk at least one time. Usually, people restrict themselves to a shape-regular bisection for the refinement. Defining
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