A Parallel Implementation of the p-Version of the Finite Element Method

Abstract

An iterative method based on the textured decomposition (TD) is developed in order to solve the systems of linear equations arising in the p-version of the finite element method. The iteration is used to implement the p-version in parallel on an MIMD computer NCUBEIsix. The objectives are twofold: to achieve high computational efficiency (that is, computational load should be balanced among the processors) and simultaneously to achieve rapid convergence. A superelement, consisting of four adjacent rectangular finite elements, is constructed for two-dimensional problems. Based on the structural property of the shape functions, each superelement is partitioned into three blocks in two different ways, and a two-leaf TD is used. Computations for a superelement associated with each leaf are assigned to two processors and are performed in parallel. A new preconditioner is introduced to accelerate convergence in a preconditioned textured decomposition (PTD). A special local communication strategy is used to avoid global assembly and global communication. Two model problems including a Laplace equation on a rectangular domain with a near singular solution and a Poisson equation on an L-shaped domain, are solved. The conjugate gradient (CG) method, the TD method, the recursive textured decomposition (RTD) method, both with and without preconditioning, and the classical iterative methods (Jacobi, Gauss–Seidel (GS), successive overrelaxation (SOR)), are used to solve both model problems. Load balance, speedup ratio, and spectral radii of the various iterations are studied The test results indicate that recursive PTD with a local communication strategy gives at least a $30\% $ improvement in computational time over the other methods.