Dynamic Load Balancing in a Lightweight Adaptive Parallel Multigrid PDE Solver

A parallel version of an adaptive multigrid solver for partial diierential equations is considered. The main emphasis is put on the load balancing algorithm to distribute the adaptive grids at runtime. The background and some applications of space-lling curves are discussed, which are later on used as the basic principle of the load-balancing heuristic. A tight integration of space-lling curves as a memory addressing scheme into the numerical algorithm is proposed. Some experiments on a cluster of PCs demonstrates the parallel eeciency and scalability of the approach. 1 An adaptive multigrid solver Our goal is to solve a partial diierential equation as fast as possible. We consider a multigrid solver, adaptive grid reenement and their eecient parallelization. We have to develop a parallel multigrid code that is almost identical to the sequential implementation. The computational workload has to be distributed into similar sized partitions and, at the same time, the communication between the processors has to be small. The underlying computer model takes into account the local processor execution time and the communication time. The rst term is proportional to the number of operations and the second one depends on the amount of data to be transferred between processors. The PDE is discretized by nite diierences. We set up the operator as a set of diierence stencils from one node to its neighboring nodes in the grid, which can be easily determined: Given a node, its neighbors can be only on a limited number of level, or one level up or down. The distance to the neighbor is determined by the level they share. So pure geometric information is suucient to apply the nite diierence operator to some vector. We avoid the storage of the stiiness matrix or any related information. For the iterative solution of the equation system, we have to implement matrix multiplication, which is to apply the operator to a given vector. A loop over all nodes in the hash table is required for this purpose. We take a strictly node-based approach. The nodes are stored in a hash table. Each interior node represents one unknown. Neither elements nor edges are stored. We use a one-irregular grid with`hanging' nodes whose values are determined by interpolation. This is equivalent to the property that there is at most onèhanging' node per edge. The one-irregular condition is a kind of a geometric smoothness condition for the adaptive grid. Additionally we …