Finite Element Preconditioning on Spectral Element Discretizations for Coupled Elliptic Equations

Finite Element Preconditioning on Spectral Element Discretizations for Coupled Elliptic Equations

On BPS type domain decomposition preconditioner for finite element discretizations of 3-d elliptic equations∗†

BPS is a well known an efficient and rather general domain decomposition DirichletDirichlet type preconditioner, suggested in the famous series of papers Bramble, Pasciak and Schatz (1986-1989). Since then, it has been serving as the origin for the whole family of domain decomposition Dirichlet-Dirichlet type preconditioners-solvers as for h so hp discretizations of elliptic problems. For its o...

Multiscale Finite Element Methods for Elliptic Equations

Here and throughout this chapter, the Einstein convention for repeated indices are assumed. The problem (9.1) a model multiscale problem which arises in the modeling of composite materials and the flow transport in heterogeneous porous media. The main difficulty in solving it by standard finite element method is that when ε is small, the underlying finite element mesh h must be much less than ε...

A posteriori error estimates for nonlinear problems. Lr-estimates for finite element discretizations of elliptic equations

— We extend the gênerai framework of [18] for deriving a posteriori error estimâtes for approximate solutions of noniinear elliptic problems such ihat it also yields L'-error estimâtes. The gênerai results are applied to finite element discretizations of scalar quasilinear elliptic pdes of 2nd order and the stationary incompressible Navier-Stokes équations. They immediately yield a posteriori e...

Two-scale Finite Element Discretizations for Partial Differential Equations ∗1)

Some two-scale finite element discretizations are introduced for a class of linear partial differential equations. Both boundary value and eigenvalue problems are studied. Based on the two-scale error resolution techniques, several two-scale finite element algorithms are proposed and analyzed. It is shown that this type of two-scale algorithms not only significantly reduces the number of degree...