Parallel MIC(0) preconditioning of 3D elliptic problems discretized by Rannacher–Turek finite elements

Abstract

Novel parallel algorithms for the solution of large FEM linear systems arising from second order elliptic partial differential equations in 3D are presented. The problem is discretized by rotated trilinear nonconforming Rannacher–Turek finite elements. The resulting symmetric positive definite system of equations A x = f is solved by the preconditioned conjugate gradient algorithm. The preconditioners employed are obtained by the modified incomplete Cholesky factorization MIC(0) of two kinds of auxiliary matrices B that both are constructed as locally optimal approximations of A in the class of M -matrices. Uniform estimates for the condition number κ ( B − 1 A ) are derived. Two parallel algorithms based on the different block structures of the related matrices B are studied. The numerical tests confirm theory in that the algorithm scales as O ( N 7 / 6 ) in the matrix order N .