Randolph E. Bank

Some a posteriori error estimators for elliptic partial differential equations

We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations. The estimators are based on solving local Neumann problems in each element. The estimators differ in how they enforce consistency of the Neumann problems. We prove that as the mesh size decreases, under suitable assumptions, two of the error estimators approach upper bounds on the norm of the true error, and all three error estimators are within multiplicative constants of the norm of the true error. We present numerical results in which one of the error estimators appears to converge to the norm of the true error.

Some errors estimates for the box method

We define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane. Our estimates show the error to be comparable to a standard Galerkin finite element method using piecewise linear polynomials.

PLTMG, a software package for solving elliptic partial differential equations : users' guide 8.0

Intended mainly for use as a reference manual, this edition encompasses all the improvements in the PLTMG software package. This book includes an internal triangle tree data structure that has simplified the internal routines.

An optimal order process for solving finite element equations

A k-level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed. The work estimate for this procedure is proportional to the number of unknowns, an optimal order result. General geometry is permitted for the underlying domain, but the shape plays a role in the rate of convergence through elliptic regularity. Finally, a short discussion of the use of this method for parabolic problems is presented.

The hierarchical basis multigrid method

SummaryWe derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

Numerical methods for semiconductor device simulation

This paper describes the numerical techniques used to solve the coupled system of nonlinear partial differential equations which model semiconductor devices. These methods have been encoded into our device simulation package which has successfully simulated complex devices in two and three space dimensions. We focus our discussion on nonlinear operator iteration, discretization and scaling procedures, and the efficient solution of the resulting nonlinear and linear algebraic equations. Our companion paper [13] discusses physical aspects of the model equations and presents results from several actual device simulations.

Global approximate Newton methods

SummaryWe derive a class of globally and quadratically converging algorithms for a system of nonlinear equations,g(u)=0, whereg is a sufficiently smooth homeomorphism. Particular attention is directed to key parameters which control the iteration. Several examples are given that have been successful in solving the coupled nonlinear PDEs which arise in semiconductor device modelling.

A posteriori error estimates based on hierarchical bases

The authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions. The authors analyze nonlinear, nonselfadjoint, and indefinite problems as well as the selfadjoint, positive-definite case. Because both the analysis and the estimator itself are quite simple, it is easy to see how various approximations affect the quality of the estimator. As examples, the authors apply the theory to some scalar elliptic equations and the Stokes system of equations.

Transient Simulation of Silicon Devices and Circuits

In this paper, we present an overview of the physical principles and numerical methods used to solve the coupled system of non-linear partial differential equations that model the transient behavior of silicon VLSI device structures. We also describe how the same techniques are applicable to circuit simulation. A composite linear multistep formula is introduced as the time-integration scheme. Newton-iterative methods are exploited to solve the nonlinear equations that arise at each time step. We also present a simple data structure for nonsymmetric matrices with symmetric nonzero structures that facilitates iterative or direct methods with substantial efficiency gains over other storage schemes. Several computational examples, including a CMOS latchup problem, are presented and discussed.

A class of iterative methods for solving saddle point problems

SummaryWe consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations.