Blockwise Adaptivity for Time Dependent Problems Based on Coarse Scale Adjoint Solutions

Abstract

We describe and test an adaptive algorithm for evolution problems that employs a sequence of “blocks” consisting of fixed, though nonuniform, space meshes. This approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing, remeshing, matrix reassembly, and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes. A major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy. We describe several strategies for achieving this goal using adjoint-based a posteriori error estimates, and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples.