A Refinement-Tree Based Partitioning Method for Adaptively Refined Grids

An adaptive multigrid method solves an elliptic partial differential equation (PDE) by beginning with a very coarse grid and cycling through phases of adaptive refinement of the grid and multigrid solution of the linear system of equations resulting from discretization of the PDE on the adaptive grid. In a parallel adaptive multigrid method, the adaptive refinement phase can cause the load balance over the processors to become unequal. If the load is too unbalanced, the grid must be repartitioned and redistributed before continuing with the solution phase. An important part of a parallel adaptive multigrid method is the method for determining this partition. In this context, it must not only produce equal sized sets to balance the load and minimize cut edges to reduce communication, but must also be very fast to not dominate the computation time of fast multigrid, and must produce similar partitions for the refinement of a grid to reduce redistribution costs. In this paper we present the K-way Refinement Tree (RTK) partitioning method, a new method for partitioning grids that were created by adaptive refinement. The method uses a weighted tree representation of the refinement process that created the grid. A traversal of the tree sums the weights of the nodes. A second traversal of the tree places subtrees into sets to quickly determine equallyweighted connected partitions.