A Comparison of Additive Schwarz Preconditioners for Parallel Adaptive Finite Elements

The Bank-Holst parallel adaptive meshing paradigm [2, 3, 1] is utilised to solve (1) in a combination of domain decomposition and adaptivity. It can be summarised as follows: Step I Mesh Partition: Starting with a coarse mesh TH , the domain is partitioned into non-overlapping subdomains: Ω = ∪pi=1Ωi. Step II Adaptive Meshing: Each processor i is provided with TH and instructed to sequentially solve the entire problem, with the stipulation that its adaptive enrichment should be limited largely to Ωi. At the end of this step, the local mesh Ti on processor i are regularised such that the global fine mesh described in Step III is conforming. Step III Global Solve: A final finite element solution is computed on the mesh Th = ∪pi=1Ti|Ωi , which is the union of the refined submeshes. An example of meshes in different steps of the Bank-Holst paradigm is illustrated in Figure 1. Discretizing (1) using linear finite elements on the global mesh Th, we arrive at the following system of linear equations: