A Parallel Algebraic Multigrid Solver For High-order Lagrange Finite Element Equations Based On HYPRE

Multigrid can be divided into geometrical multigrid (GMG) and algebraic multigrid (AMG). Comparing with GMG, AMG can be applied to more general cases and possesses stronger robustness, which is one of the most efficient methods to solve large scale scientific computation in engineering, in particular, for discretizations of partial differential equations. HYPRE is a popular software library for solving large sparse linear systems on massively parallel computers. The library is created with the primary goal of providing users with advanced parallel solvers or preconditioners, c.f. BoomerAMG. In this paper, we discuss the parallel AMG solver for high-order Lagrange finite element equations of 3D elliptic boundary problem by using HYPRE. The primary pursuits are as follows:Firstly, we introduce the HYPRE library, then describe some classic grid coarsening algorithms (e.g. RS and CLJP coarsening) and a classic parallel grid coarsening algorithm: Falgout coarsening. We also introduce the convergence theory of the MSSC which is developing in recent years.Secondly, based on subdomain partitions for high-order hierarchic finite element discretizations and by introducing a group of sub-matrixes and sub-loading vectors relative to faces, edges and corner points on each processor, we design a parallel algorithm of generating stiffness matrix and loading vector. Additionally, we adopt a reasonable order for the hierarchic bases, which not only brings convenience for programming but also improves the relaxation efficiency of parallel AMG. Numerical experiments confirm that the parallel algorithm enlarges the scale of generating stiffness matrix and has better scalability.Thirdly, based on auxiliary variational problems for higher-order finite element discretizations, we design a new AMG (so-called X-AMG) and prove that the convergence rate of X-AMG is independent of the mesh size by using the theory concerning method of successive subspace corrections, which can also be confirmed by the resulting of numerical experiments. Then we design two parallel algorithms for X-AMG. The first one, called X-AMG-I, is designed for a serial structure of stiffness matrix Although X-AMG-I is stable on the number of iteration, there exists the following faults: it is too frequent for the transformation between parallel vector and serial vector, and the efficiency of the algorithm is dependent 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 AMG is shown, by numerical experiments, to be much more efficient than BomerAMG which is directly applied to the high-order finite element matrix.