Parallel direct solvers for adaptive Finite Element and Finite Difference Methods

In this presentation the parallel multi-frontal direct solver for adaptive finite element or finite difference methods is described. The methodology is first illustrated on a simple 1D example, concerning the 1D heat transfer problem solved by the finite difference method. The execution of the solver algorithm starts with the generation of the binary elimination tree. The algorithm of the parallel solver is presented as an execution of several atomic tasks. These atomic tasks create multiple frontal matrices at leaves of the elimination tree, and then recursively execute the merging and the elimination process, up to the root of the tree. Finaly, the atomic tasks representing recursive backward substitutions are exexcuted. Having the sequence of named atomic tasks, we can create the model of concurrency, by deciding which atomic tasks can be executed in concurrent. The sequence of atomic tasks is divited into sub-sets. Each sub-set contains atomic tasks to be excuted at the same time. The execution of all tasks from a sub-set is followed by global synchronization. The concurrent multi- frontal direct solver created in this way has been implemented on the shared memory multi-core NVIDIA CUDA graphic card, with logarithmic performance.

The methodology is than extended to the finite element method. It is done by replacing only one atomic task, generating frontal matrices at the leaves of the elimination tree. Finally, the methodology is also extended for the distributed memory architecture, for adaptive finite element method computations. The atomic tasks are agglomerated into groups of tasks, associated with initial mesh elements. The computational mesh is partitioned on the level of initial mesh elements into multiple sub-domains. Each sub-domain is assigned to a single processor. The parallel multi-frontal direct solver algorithm obtained in this way is tested on 3D Direct Current (DC) borehole resistivity measurements simulations problem in deviated well. From the experiments performed on LONESTAR linux cluster it follows that the solver outperforms the MUMPS parallel solver, being known as one the best available solvers.