一種基于HYPRE的高次Lagrange有限元方程的并行代數(shù)多層網(wǎng)格法.pdf

1、湘潭大學碩士學位論文一種基于HYPRE的高次Lagrange有限元方程的并行代數(shù)多層網(wǎng)格法姓名：曹放申請學位級別：碩士專業(yè)：計算數(shù)學指導教師：舒適20080415AbstractMultigrid can be divided into geometrical multigrid (GMG) and algebraic multigrid(AMG). Comparing with GMG, AMG can be applied to m

2、ore general cases and possessesstronger robustness, which is one of the most efficient methods to solve large scale scientificcomputation in engineering, in particular, for discretizations of partial differential equatio

3、ns.HYPRE is a popular software library for solving large sparse linear systems on massivelyparallel computers. The library is created with the primary goal of providing users withadvanced parallel solvers or precondition

4、ers, c.f. BoomerAMG. In this paper, we discuss theparallel AMG solver for high-order Lagrange finite element equations of 3D elliptic boundaryproblem by using HYPRE. The primary pursuits are as follows:Firstly, we introd

5、uce the HYPRE library, then describe some classic grid coarseningalgorithms (e.g. RS and CLJP coarsening) and a classic parallel grid coarsening algorithm:Falgout coarsening. We also introduce the convergence theory of t

6、he MSSC which is devel-oping in recent years.Secondly, based on subdomain partitions for high-order hierarchic finite element dis-cretizations and by introducing a group of sub-matrixes and sub-loading vectors relativeto

7、 faces, edges and corner points on each processor, we design a parallel algorithm of gen-erating stiffness matrix and loading vector. Additionally, we adopt a reasonable order forthe hierarchic bases, which not only brin

8、gs convenience for programming but also improvesthe relaxation efficiency of parallel AMG. Numerical experiments confirm that the parallelalgorithm enlarges the scale of generating stiffness matrix and has better scalabi

9、lity.Thirdly, based on auxiliary variational problems for higher-order finite element dis-cretizations, we design a new AMG (so-called X-AMG) and prove that the convergencerate of X-AMG is independent of the mesh size by

10、 using the theory concerning method ofsuccessive subspace corrections, which can also be confirmed by the resulting of numericalexperiments. Then we design two parallel algorithms for X-AMG. The first one, called X-AMG-I

11、, is designed for a serial structure of stiffness matrix. Although X-AMG-I is stableon the number of iteration, there exists the following faults: it is too frequent for the trans-formation between parallel vector and se

12、rial vector, and the efficiency of the algorithm isdependent on the smoother closely. Thus we design the second parallel algorithm for X-AMG, called X-AMG-II, which improves the X-AMG-I. The resulting new parallel AMGis