Engineering topics

Shapira, Yair

Publications and source records attributed to Shapira, Yair.

3 records

NASA NTRS1996

Multigrid Techniques for Highly Indefinite Equations

A multigrid method for the solution of finite difference approximations of elliptic PDE's is introduced. A parallelizable version of it, suitable for two and multi level analysis, is also defined, and serves as a theoretical tool for deriving a suitable implementation for the main version. For indefinite Helmholtz equations, this analysis provides a suitable mesh size for the coarsest grid used. Numerical experiments show that the method is applicable to diffusion equations with discontinuous coefficients and highly indefinite Helmholtz equations.

Shapira, Yair↗

NASA NTRS1993

An automatic multigrid method for the solution of sparse linear systems

An automatic version of the multigrid method for the solution of linear systems arising from the discretization of elliptic PDE's is presented. This version is based on the structure of the algebraic system solely, and does not use the original partial differential operator. Numerical experiments show that for the Poisson equation the rate of convergence of our method is equal to that of classical multigrid methods. Moreover, the method is robust in the sense that its high rate of convergence is conserved for other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. No double discretization or special treatment of sub-domains (e.g. boundaries) is needed. When supplemented with a vector extrapolation method, high rates of convergence are achieved also for anisotropic and discontinuous problems and also for indefinite Helmholtz equations. A new double discretization strategy is proposed for finite and spectral element schemes and is found better than known strategies.

Shapira, Yair↗

NASA NTRS1992

Upper bounds for convergence rates of vector extrapolation methods on linear systems with initial iterations

The application of the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) to a vector sequence obtained by the linear iterative technique x(sub j) + 1 = Ax(sub j) = b,j = 1,2,..., is considered. Both methods produce a two dimensional array of approximations s(sub n,k) to the solution of the system (I - A)x = b. Here, s(sub n,k) is obtained from the vectors x(sub j), n is less than or equal to j is less than or equal to n + k + 1. It was observed in an earlier publication by the first author that the sequence s(sub n,k), k = 1,2,..., for n greater than 0, but fixed, possesses better convergence properties than the sequence s(sub 0,k), k = 1,2,.... A detailed theoretical explanation for this phenomenon is provided in the present work. This explanation is heavily based on approximations by incomplete polynomials. It is demonstrated by numerical examples when the matrix A is sparse that cycling with s(sub n,k) for n greater than 0, but fixed, produces better convergence rates and costs less computationally than cycling with s(sub 0,k). It is also illustrated numerically with a convection-diffusion problem that the former may produce excellent results where the latter may fail completely. As has been shown in an earlier publication, the results produced by s(sub 0,k) are identical to the corresponding results obtained by applying the Arnoldi method or generalized minimal residual scheme (GMRES) to the system (I - A)x = b.

Sidi, Avram↗