Fast Solving Algorithms For Discrete Models Of Two Types Of Lattice Block Materials

In this paper, we propose two kinds of preconditioning methods for the discretemodels on lattice block materials with q = 3 and q = 4 reference nodes,respectively. One is the block preconditioned conjugate gradiet(BPCG) algorithmwith a block diagonal inverse as a preconditioner. The numerical resultshave been shown that the BPCG method is of good efficiency and robustness applyingto the solution of the lattice block material with q = 3, but for honeycomblattices (q = 4), especially whenαis very small, the efficiency of this methodbecomes poor. To deal with such situations, we propose another preconditioningGMRES method (PGMRES) based on the block lower triangular inverse as a preconditioner.Whenαis very small, compared with the BPCG and the PGMRESwith a block diagonal preconditioner, this PGMRES method for the honeycomblattice(q = 4) discrete model is best both in the calculation of the stability andCPU times. In both preconditioning methods,in order to improve the efficiencyof the inner iterations, we have made use of PCG method with GAMG preconditionerbased on scalar elliptic problems to finding the inverse of each sub-blockmatrix. At the same time, internal iteration counts and the condition numbersof each sub-bloch matrix before and after preconditioning using GAMG methodare presented and it is shown that the PCG method with GAMG preconditionerhas better efficiency and robustness. In addition, for the lattice block materialwith q = 3, we use the Taylor's expansion to establish the limiting continuous(PDEs) equations of three typical reference nodes. Theoretical support for thereasonableness of the inner iterations is partly presented by making the analysisof H₁⁰(Ω) elliptical condition for the sub-systems in the inner iterations.