Several Algebraic Multigrid Methods And Domain Decomposition Methods For Solving Radiative Difusion Problem And Linear Elasticity Problem

Algebraic multigrid (AMG) method and domain decomposition method(DDM) are two fast algorithms which are widely used for solving large-scale linear systems arising from the partial diferential equations (PDEs)discretization. For discretized systems of complicated PDEs, there are stillmany problems to be studied further for the two methods. In this disser-tation, we focus on the efcient (parallel) AMG and DDM research for twomodels with wide application background, the main work is as follows.The performance of preconditioner B01is tested and analyzed for sometypical linear systems discretizated from2D3T equations. By introducingseveral factors which are able to describe coupling relationship of the threetemperatures and diagonally dominance of the diagonal sub-matrices, wepropose a new preconditioner B2, in which Block Diagonal-type and PCTL-type preconditioners are selected adaptively. Numerical results of the test-ing for some data extracted from LARED-S show that compared with theRSAMG-preconditioned and the B01-preconditioned GMRES methods, theB2-preconditioned GMRES is more robust and of higher efciency. Further-more, the parallel algorithm of Bp2(B2) for JASMIN data interface is alsoimplemented by using the strategy of processor-grouping. For some modelsof practical application background, numerical results show that comparedwith the BoomerAMG-preconditioned GMRES solver, our parallel solver isof better algorithm scalability and faster. To be more user-friendly, we ad-ditionally provide a serial interface for Bp2.Two discrete schemes–mixed five-point scheme and SFVE scheme onSAMR for a2D stationary difusion problem are firstly discussed, the cor-relation between the behavior of solution function near the coarse-fine in-terface, the interpolation accuracy and the approximation of the numericalsolution is then analyzed. The SFVE scheme proves to be more general.We design a two-level (TL) method for the two schemes on SAMR, and alsoprove the uniform convergence of TL method for the SFVE scheme. Further- more, for a2D3T model, we obtain the corresponding adaptive PCTL pre-conditioner Btl2by replacing RSAMG with TL method for the sub-systems.One advantage of this preconditioner is that it is easy to be implementedin JASMIN. For the strong-coupling case, numerical results show that theBtl2-preconditioned GMRES is more robust and more efcient compared withthe RSAMG-preconditioned GMRES.A substructuring preconditioner with simple coarse spaces is constructedto solve the preserving-symmetry finite volume element (SFVE) discretizedsystem for a two-dimensional radiative difusion equation. It needs to solvetwo kinds of subproblems which are self-similar to the original problem. Nu-merical results show that the iteration number of the corresponding PCGdepends weakly on the mesh size. Furthermore, by introducing an auxiliarylinear finite element discretized system, we prove that the condition numberof the preconditioned system is nearly optimal(O(log3dh)) for a simplifiedmodel.The hierarchical quadratic finite element discretizations on some3Dtypical local anisotropic grids are discussed. By constructing special blocksmoothers which are able to efectively eliminate the high-frequency error dueto the anisotropy of the grids, we develop a type of two-level method whoseconvergence depends weakly on the mesh size. And then, we obtain thecorresponding multi-level method by applying the existing AMG (DAMG orDAMG-CG) methods which are applicable to problems on anisotropic gridsto the solution of the coarse level (linear element) equations. Furthermore,the resulting multi-level method is applied to a3D practical example. Thenumerical results verify the high efciency and robustness of the proposedalgorithms.