Petter E. Bjørstad

Progress in Sparse Matrix Methods for Large Linear Systems On Vector Supercomputers

This paper summarizes progress in the use of direct methods for solving very large sparse symmetric positive definite systems of linear equations on vector supercomputers. Sparse di rect solvers based on the multifrontal method or the general sparse method now outperform band or envelope solvers on vector supercomputers such as the CRAY X-MP. This departure from conventional wisdom is due to several advances. The hardware gather/scatter feature or indirect address feature of some recent vector super computers permits vectorization of the general sparse factorization. Other advances are algo rithmic. The new multiple minimum degree algo rithm calculates a powerful ordering much faster than its predecessors. Exploiting the supernode structure of the factored matrix provides vectori zation over nested loops, giving greater speed in the factorization module for the multifrontal and general sparse methods. Out-of-core versions of both methods are now available. Numerical re sults on the CRAY X-MP for several structural engineering problems demonstrate the impact of these improvements.

Fast Numerical Solution of the Biharmonic Dirichlet Problem on Rectangles

A new method for the numerical solution of the first biharmonic Dirichlet problem in a rectangular domain is presented. For an

On the spectra of sums of orthogonal projections with applications to parallel computing

N \times N

To overlap or not to overlap: a note on a domain decomposition method for elliptic problems

mesh the complexity of this algorithm is on the order of

SOLVING ELLIPTIC PROBLEMS ON REGIONS PARTITIONED INTO SUBSTRUCTURES

N^2

Efficient matrix multiplication on SIMD computers

arithmetic operations. Only one array of order

Additive Schwarz Methods for Elliptic Mortar Finite Element Problems

N \times N

Conformal mapping of circular arc polygons

mesh the complexity of this algorithm is on the order of

High precision solutions of two fourth order eigenvalue problems

N^2

To Overlap or Not to Overlap

and a workspace of size less than