An efficient direct parallel spectral-element solver for separable elliptic problems

An efficient direct parallel elliptic solver based on the spectral element discretization is developed. The direct solver is based on a matrix decomposition approach which reduces multi-dimensional separable problems to a sequence of one-dimensional problems that can be efficiently handled by a static condensation process. Thanks to the spectral accuracy and the localized nature of a spectral element discretization, this elliptic solver is spectrally accurate and can be efficiently parallelized, and it can serve as an essential building block for large scale high-performance solvers in computational fluid dynamics and computational materials science.

[1]  G. Karniadakis,et al.  Spectral/hp Element Methods for CFD , 1999 .

[2]  Jie Shen,et al.  Efficient Spectral-Galerkin Methods III: Polar and Cylindrical Geometries , 1997, SIAM J. Sci. Comput..

[3]  Osman A. Basaran,et al.  Dynamics and breakup of a contracting liquid filament , 2004, Journal of Fluid Mechanics.

[4]  Christine Bernardi,et al.  Discr'etisations variationnelles de probl`emes aux limites elliptiques , 2004 .

[5]  W. Couzy,et al.  A fast Schur complement method for the spectral element discretization of the incompressible Navier-Stokes equations , 1995 .

[6]  Jie Shen,et al.  Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials , 1994, SIAM J. Sci. Comput..

[7]  Jack Dongarra,et al.  ScaLAPACK Users' Guide , 1987 .

[8]  Jie Shen,et al.  E-cient Chebyshev-Legendre Galerkin Methods for Elliptic Problems , 1996 .

[9]  John R. Rice,et al.  Direct solution of partial difference equations by tensor product methods , 1964 .

[10]  Michel Deville,et al.  Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation , 1984 .

[11]  I. Babuska,et al.  Efficient preconditioning for the p -version finite element method in two dimensions , 1991 .

[12]  Jie Shen,et al.  Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method , 2006, J. Comput. Phys..

[13]  H.M. Tufo,et al.  Terascale Spectral Element Algorithms and Implementations , 1999, ACM/IEEE SC 1999 Conference (SC'99).

[14]  Jie Shen,et al.  A new class of truly consistent splitting schemes for incompressible flows , 2003 .

[15]  Jie Shen,et al.  Efficient Spectral-Galerkin Method II. Direct Solvers of Second- and Fourth-Order Equations Using Chebyshev Polynomials , 1995, SIAM J. Sci. Comput..

[16]  Anthony T. Patera,et al.  Fast direct Poisson solvers for high-order finite element discretizations in rectangularly decomposable domains , 1986 .

[17]  G. Karniadakis,et al.  Spectral/hp Element Methods for Computational Fluid Dynamics , 2005 .

[18]  Olof B. Widlund,et al.  Iterative Substructuring Methods for Spectral Element Discretizations of Elliptic Systems I: Compressible Linear Elasticity , 1999, SIAM J. Numer. Anal..

[19]  Gene H. Golub,et al.  On direct methods for solving Poisson's equation , 1970, Milestones in Matrix Computation.

[20]  Dale B. Haidvogel,et al.  The Accurate Solution of Poisson's Equation by Expansion in Chebyshev Polynomials , 1979 .

[21]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[22]  P. Fischer,et al.  High-Order Methods for Incompressible Fluid Flow , 2002 .

[23]  Petter E. Bjørstad,et al.  Timely Communication: Efficient Algorithms for Solving a Fourth-Order Equation with the Spectral-Galerkin Method , 1997, SIAM J. Sci. Comput..