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 Ax = 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 .

Parallel Performance Evaluation of MIC(0) Preconditioning Algorithm for Voxel μFE Simulation

“…A preconditioning algorithm was developed using a parallel MIC(0) elasticity solver [20], based on a parallel MIC(0) solver for the scalar elliptic problem [3]. The preconditioner uses the isotropic variant of the displacement decomposition (DD) [6,10].…”

“…Here, diagonal entries of B e are modified to hold the row-sum criteria (for more details see [3]). Assembling the locally defined matrices B e we obtain the global matrix B = e∈ω h R T e B e R e .…”

“…Assembling the locally defined matrices B e we obtain the global matrix B = e∈ω h R T e B e R e . The condition number estimate κ(B −1 A) ≤ 3 holds uniformly with respect to the mesh parameter and to possible coefficient jumps (for the related analysis see discussion presented in [3]). After this modification we obtain matrix B with its diagonal blocks (corresponding to horizontal cross sections) being diagonal matrices.…”

A general approach to the construction of nonconforming finite elements on convex polytopes

“…. , f d−1 needed in (3) in the case of an orthogonal parallelepiped can be generated from just one single function f ∈ C[0, 1]. It will turn out that the remainder R tr (f) of the trapezoidal rule…”

“…For such polytopes, the extension of P 1 to R(K) has often been achieved by adding quadratic functions such as [1][2][3][4][5][6][7][8][9][11][12][13][14][15][16][17]. More generally, we study if f 1 , .…”