A parallel solver for the hp-version of finite element methods

A prototype software system has been implemented for the parallel solution of linear systems associated with the hp-version, using the master/slave model and the Parallel Virtual Machine (PVM) message passing library. In a series of performance tests, significant speed-up was achieved in those typical cases for the hp-version where there was sufficient computational granularity to justify use of the parallel method. These tests indicate that the algorithms devised for element ordering, load distribution and load balancing are sufficiently robust. These tests also indicate that, even though communication overhead in a network environment is relatively high, there is significant potential for scaling the method to larger processor ensembles. The solution method developed was implemented as a four step procedure. In step 1, a dynamic load distribution algorithm selects an element for each idle process based on the mesh topology. In step 2, element descriptions are communicated to each process where local matrix computations are performed in parallel. In step 3, regional assembly and elimination operations are carried out in parallel to produce multiple 'frontal matrices' which are eventually returned to a host process for final solution. In step 4, the solution to the global stiffness matrix is back-substituted into each previous frontal matrix to determine the values of all unknowns eliminated during the parallel assembly procedure.