Numerical simulations of viscoelastic flow problems require the solution of large, typically sparse, systems of equations which inherit the (highly) nonlinear coupling of the original PDE model. Recent approaches, such as the elastic viscous split stress (EVSS) methods, while allowing more flexibility and the treatment of complex rheological models come at the cost of increased problem size. At...

In this report, we discuss block preconditioners used to solve the incompressible Navier-Stokes equations. We emphasize on the approximation of the Schur complement used in SIMPLE-type preconditioners. In the usual formulation, the Schur complement uses scaling with the diagonal of the convection diffusion matrix. A variant of SIMPLE, SIMPLER is studied. Convergence of the SIMPLER preconditione...

In this paper, we discuss various techniques for solving the system of linear equations that arise from the discretization of the incompressible Stokes equations by the finite-element method. The proposed solution methods, based on a suitable approximation of the Schur-complement matrix, are shown to be very effective for a variety of problems. In this paper, we discuss three types of iterative...

We present a class of parallel preconditioning strategies utilizing multilevel block incomplete LU (ILU) factorization techniques to solve large sparse linear systems. The preconditioners are constructed by exploiting the concept of block independent sets (BISs). Two algorithms for constructing BISs of a sparse matrix in a distributed environment are proposed. We compare a few implementations o...

A Legendre spectral collocation method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approximation of the Laplacia...

In this work, we propose a preconditioned GMRES solver for a Schur complement equation of the linearized fluid-structure interaction problem, with respect to the displacement unknowns only on the interface. The preconditioning for the Schur complement equation requires approximate solutions of the fluid and structure sub-problems with appropriate boundary conditions on the interface, in particu...

Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, secondorder cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of th...

We propose a primal-dual path-following Mehrotra-type predictor-corrector method for solving convex quadratic semidefinite programming (QSDP) problems. For the special case when the quadratic term has the form 1 2 X •(UXU), we compute the search direction at each iteration from the Schur complement equation. We are able to solve the Schur complement equation efficiently via the preconditioned s...

We consider a linear-quadratic elliptic control problem (LQECP). For the problem we consider here, the control variable corresponds to the Neumann data on the boundary of a convex polygonal domain. The optimal control unknown is the one for which the harmonic extension approximates best a specified target in the interior of the domain. We propose multilevel preconditioners for the reduced Hessi...

A dual logarithmic barrier method for solving large, sparse semidefinite programs is proposed in this paper. The method avoids any explicit use of the primal variable X and therefore is well-suited to problems with a sparse dual matrix S. It relies on inexact Newton steps in dual space which are computed by the conjugate gradient method applied to the Schur complement of the reduced KKT system....