Multigrid methods and parallel computations for elliptic problems, with an emphasis on linear elasticity

Multigrid methods on elliptic problems are developed and numerical computations using a parallel computer are given.;First, for second-order elliptic boundary value problems, a nonconforming multigrid method using the coarser-grid correction on the conforming finite element subspaces is developed. The convergence proof with an arbitrary number of smoothing steps for V-cycle is presented and numerical results are included. Second, a conforming P-1 mixed finite element method using the method of reduced integration is developed for the two-dimensional pure traction boundary value problem of linear elasticity. The convergence is uniform as the material becomes nearly incompressible. In addition, a multigrid method using this finite element is developed. Also the convergence is uniform even as the material becomes nearly incompressible. A heuristic argument for acceleration of the multigrid method is discussed as well. Numerical results with and without this acceleration as well as performance estimates on a parallel computer are included.;Third, the implementations of domain decomposition method, SOR, multigrid method and conjugate gradient method on the CM-5 and the Cray C90 are described for Laplace's equation on the unit square and an L-shaped region. The domain decomposition method uses the Schwarz alternating method. On the CM-5 the V-cycle multigrid with symmetric smoothings on the P-1 finite element spaces is run with red/black Gauss-Seidel relaxation. The multigrid with natural order Gauss-Seidel relaxation is used on the Cray C90. While natural order SOR is used on the Cray C90, R/B SOR is performed on the CM-5. The multigrid is the fastest method on the CM-5, and three methods except SOR give similar performances on the Cray C90.