A Domain Decomposition for a Parallel Adaptive Meshing Algorithm

We describe a domain decomposition algorithm for use in the parallel adaptive meshing paradigm of Bank and Holst. Our algorithm has low communication , makes extensive use of existing sequential solvers, and exploits in several important ways data generated as part of the adaptive meshing paradigm. Numerical examples illustrate the effectiveness of the procedure. In [4, 5], we introduced a general approach to parallel adaptive meshing for systems of elliptic partial differential equations. This approach was motivated by the desire to keep communications costs low, and to allow sequential adap-tive software (such as the software package pltmg used in this work) to be employed without extensive recoding. Our discussion is framed in terms of continuous piecewise linear triangular finite element approximations used in pltmg, although most ideas generalize to other approximation schemes. Our original paradigm, called Plan A in this work, has three main components: Step I: Load Balancing. We solve a small problem on a coarse mesh, and use a posteriori error estimates to partition the mesh. Each subregion has approximately the same error, although subregions may vary considerably in terms of numbers of elements or gridpoints. Step II: Adaptive Meshing. Each processor is provided the complete coarse mesh and instructed to sequentially solve the entire problem, with the stipulation that its adaptive refinement should be limited largely to