S. H. Lui

Computation of Pseudospectra by Continuation

The concept of pseudospectrum was introduced by Trefethen to explain the behavior of nonnormal operators. Many phenomena (for example, hydrodynamic instability and convergence of iterative methods for linear systems) cannot be accounted for by eigenvalue analysis but are more understandable by examining the pseudospectra. The straightforward way to compute pseudospectra involves many applications of the singular value decomposition (SVD). This paper presents several fast continuation methods to calculate pseudospectra of different types of matrices.

On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs

SummaryThe Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain.In this paper, proofs of convergence of some Schwarz Alternating Methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well some coupled nonlinear PDEs are shown to converge to some solution on finitely many subdomains, even when multiple solutions are possible. In the coupled system case, each subdomain PDE is linear, decoupled and can be solved concurrently with other subdomain PDEs. These results are applicable to several models in population biology.

On Schwarz Alternating Methods for the Incompressible Navier--Stokes Equations

The Schwarz alternating method can be used to solve linear elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which result from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers four Schwarz alternating methods for the N-dimensional, steady, viscous, incompressible Navier--Stokes equations,

On Schwarz Alternating Methods for Nonlinear Elliptic PDEs

N \leq 4

Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

. It is shown that the Schwarz sequences converge to the true solution provided that the Reynolds number is sufficiently small.

Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers several Schwarz alternating methods for nonlinear elliptic problems. We show that Schwarz alternating methods can be embedded in the framework of common techniques such as Banach and Schauder fixed point methods and global inversion methods used to solve these nonlinear problems.

Legendre spectral collocation in space and time for PDEs

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump.

A pseudospectral mapping theorem

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence. In the literature, the analysis of optimized Schwarz methods relies on Fourier analysis and so the domains are restricted to be regular (rectangular). In this paper, we express the interface operator of an optimized Schwarz method in terms of Poincare-Steklov operators. This enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1-O(h^1^/^4) on a class of general domains, where h is the discretization parameter. This is the predicted rate for a second order optimized Schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations.

Adaptive Wavelet Schwarz Methods for the Navier-Stokes Equation

Spectral methods solve partial differential equations numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods which converge spectrally in both space and time have appeared recently. This paper shows that a Legendre spectral collocation method of Tang and Xu for the heat equation converges exponentially quickly when the solution is analytic. We also derive a condition number estimate of the method. Another space-time spectral scheme which is easier to implement is proposed. Numerical experiments verify the theoretical results.

On accelerated convergence of nonoverlapping Schwarz methods

The pseudospectrum has become an important quantity for analyzing stability of nonnormal systems. In this paper, we prove a mapping theorem for pseudospectra, extending an earlier result of Trefethen. Our result consists of two relations that are sharp and contains the spectral mapping theorem as a special case. Necessary and sufficient conditions for these relations to collapse to an equality are demonstrated. The theory is valid for bounded linear operators on Banach spaces. For normal matrices, a special version of the pseudospectral mapping theorem is also shown to be sharp. Some numerical examples illustrate the theory.