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SIAM J. on Numerical Analysis

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1986

Volume 23, Issue 6, pp. 1097-1302


Iterative Methods for the Solution of Elliptic Problems on Regions Partitioned into Substructures

Petter E. Bjørstad and Olof B. Widlund

SIAM J. Numer. Anal. 23, pp. 1097-1120 (24 pages) | Cited 118 times

Online Publication Date: July 14, 2006

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Finite element problems can often naturally be divided into subproblems which correspond to subregions into which the region has been partitioned or from which it was originally assembled. A class of iterative methods is discussed in which these subproblems are solved by direct methods, while the interaction across the curves or surfaces which divide the region is handled by a conjugate gradient method. A mathematical framework for this work is provided by regularity theory for elliptic finite element problems and by block Gaussian elimination. A full development of the theory, which shows that certain of these methods are optimal, is given for Lagrangian finite element approximations of second order linear elliptic problems in the plane. Results from numerical experiments are also reported.

A Proof of Convergence of Gummel’s Algorithm for Realistic Device Geometries

Thomas Kerkhoven

SIAM J. Numer. Anal. 23, pp. 1121-1137 (17 pages) | Cited 13 times

Online Publication Date: July 14, 2006

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Convergence is proven for a typical example from the class of highly successful decoupling algorithms for semiconductor simulation, collectively known as Gummel’s method, for one-, two- and three-dimensional models. Because a nonlinear equation is solved for the potential $u$ at every step, the considered version corresponds closely to the algorithms used for numerical computation in practice. As opposed to most earlier publications, the dependence of the regularity of the solution on the device geometry and the nature of the boundary conditions for the system of mixed boundary value problems is considered.
By a detailed analysis of the boundary conditions for a typical two- or three-dimensional device a theoretical regularity result by Murthy and Stampacchia can be applied. Hence it follows that for a physically realistic device geometry the solution is sufficiently regular for the algorithm to converge.

Analysis of Chebyshev Collocation Methods for Parabolic Equations

N. Bressan and A. Quarteroni

SIAM J. Numer. Anal. 23, pp. 1138-1154 (17 pages) | Cited 6 times

Online Publication Date: July 14, 2006

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Time discretizations of both the heat equation and the advection-diffusion equation in two space variables are analyzed. For space approximation the pseudospectral Chebyshev method, which enjoys infinite order of accuracy for smooth solutions, is used. The equation is collocated at the Gauss–Chebyshev points, and derivatives are computed by the pseudospectral differencing technique.
At any time interval the pseudospectral solution, which is a polynomial of degree $N$, is advanced in time using the implicit $\theta $-method for the diffusive part of the equation, while the advective term is dealt with explicitly.
We prove unconditional stability and optimal error bounds, depending on both $N$ and $\Delta t$ (the time-step), in the norms of the weighted Sobolev spaces.
The method considered here is the most commonly used spectral method for parabolic equations. To solve the linear system arising at any time interval, efficient iterative techniques with scaling are presented.

Error Estimates for a Numerical Method for an Ill-Posed Cauchy Problem for the Heat Equation

Peter Monk

SIAM J. Numer. Anal. 23, pp. 1155-1172 (18 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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Logarithmic convexity type continuous dependence results for discrete solutions of the heat equation are proved. Using these results, error estimates for a least squares penalty method for approximating a noncharacteristic Cauchy problem for the heat equation are obtained. Numerical results are presented.

Simplified Godunov Schemes for $2 \times 2$ Systems of Conservation Laws

J. P. Vila

SIAM J. Numer. Anal. 23, pp. 1173-1192 (20 pages) | Cited 3 times

Online Publication Date: July 14, 2006

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Some Godunov type schemes for solving Isentropic Gas Dynamic equations are presented. After analyzing a one intermediary state scheme and its properties, we prove consistency with entropy condition for time continuous limit solutions of a simplified Godunov scheme by techniques due to Osher. We prove also similar results for high order versions of these numerical schemes.

The Time Evolution of Spectral Discretizations of Hyperbolic Systems

Liviu Lustman

SIAM J. Numer. Anal. 23, pp. 1193-1198 (6 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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A Chebyshev collocation spectral method for hyperbolic systems is considered, particularly for those initial boundary problems which admit only solutions tending to zero at large times. It is shown that the numerical solution will also vanish at infinity, under the same conditions. These results are then generalized for other spectral approximations.

On Fornberg’s Numerical Method for Conformal Mapping

Rudolf Wegmann

SIAM J. Numer. Anal. 23, pp. 1199-1213 (15 pages) | Cited 4 times

Online Publication Date: July 14, 2006

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It is shown that Fornberg’s method [1] is a discretized version of an iterative process which is essentially the same we introduced and studied in [7], [8]. The main difference consists in the determination of the shift $U_k $ in each iterative step. While we give an explicit representation of $U_k $, Fornberg determines $U_k $ from a linear equation, which involves a certain integral operator $R$. We prove some properties and estimates for this operator which explain several experimental results reported in [1], especially the fact that the conjugate gradient method is so efficient. The results of [8] concerning the convergence can be applied also for Fornberg’s method. Finally, we give estimates for the numerical error.

Superconvergent Approximations to the Solution of a Boundary Integral Equation on Polygonal Domains

G. A. Chandler

SIAM J. Numer. Anal. 23, pp. 1214-1229 (16 pages) | Cited 5 times

Online Publication Date: July 14, 2006

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The solution to the interior Dirichlet problem for Laplace’s equation in a two-dimensional domain is the double layer potential of a double layer distribution satisfying a second kind boundary integral equation. This may be solved numerically by Galerkin’s method using piecewise polynomials of degree $r$. The iterated Galerkin solution may then be calculated, which gives an order $2r + 2$ approximation if the domain is smooth. However if the domain is polygonal, the corners cause singularities which degrade the order of convergence. Here it is shown that a suitable grading of the mesh near the corners almost restores the rate of convergence when the error is measured in the uniform norm. We then describe a second means of calculating a higher order approximation from the Galerkin solution. This is cheaper to calculate than the iterated Galerkin solution, but maintains the same order of convergence.

Bivariational Methods for Linear Integral Equations with Nonsymmetric Kernels

P. D. Robinson and P. K. Yuen

SIAM J. Numer. Anal. 23, pp. 1230-1240 (11 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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Bivariational methods are used to develop upper and lower bounding functionals for arbitrary inner products $\langle {g,\phi } \rangle $ associated with the solutions $\phi $ of a class of linear integral equations with nonsymmetric kernels. The underlying structure of the variational principles is seen to relate to a linear combination of the integral equation and an adjoint equation. The new functionals derived are more accurate than previous bounds obtained from a direct approach. Their efficiency is illustrated by using them to calculate actual pointwise bounds on some solutions of integral equations. In two test cases, the accuracy achieved was superior to that arising from a standard 160-step Simpson quadrature calculation.

A Conservative Uniformly Accurate Difference Method for a Singular Perturbation Problem in Conservation Form

Alan E. Berger

SIAM J. Numer. Anal. 23, pp. 1241-1253 (13 pages) | Cited 8 times

Online Publication Date: July 14, 2006

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A conservative three point finite difference method is presented for the numerical solution of the singular perturbation problem $\varepsilon u_{xx} + (b(x)u)_x = f(x)$, $0 < x < 1$, $u(0)$ and $u(1)$ given, $b(x) \ne 0$. Certain a priori estimates are established for this problem and are then used to obtain uniform error estimates for the difference scheme. Some illustrative numerical results are also given.

Asymptotic Expansions for the Midpoint Rule Applied to Delay Differential Equations

Maarten de Gee

SIAM J. Numer. Anal. 23, pp. 1254-1272 (19 pages)

Online Publication Date: July 14, 2006

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Let $u^ * $ be the solution of the delay differential equation \[ \begin{gathered} u'(t) = f(t,u(t),u(t - \tau ))\quad {\text{for}}\,t > 0, \hfill \\ u(t) = \varphi (t)\qquad{\text{for}} - \tau \leqq t \leqq 0. \hfill \\ \end{gathered} \]For $t_j = jh$, let $u_h (t_j )$ be the approximation for $u^ * (t_j )$ obtained by the explicit midpoint rule, initialized by one Euler step. It is shown that if $f$ and $\varphi $ are sufficiently differentiable and ${\tau / {h \in \mathbb{N}}}$, there are functions $e_k $ such that for $j$ even, \[u_h \left( {t_j } \right) - u^ * \left( {t_j } \right) = \sum _{k = 1}^r {h^{2k} e_k } \left( {t_j } \right) + O\left( {h^{2r + 2} } \right).\] This expansion of the global discretization error exists in spite of the fact that the solution $u^ * $ usually has jump discontinuities in its derivatives at the points $t = n\tau $, $n \in \mathbb{N}$. Thus repeated extrapolation to the limit (Richardson extrapolation) may be applied to improve the approximation. Furthermore, it is shown that if a set of fixed stepsizes $h_i $ is used, then one may apply local extrapolation as in the Gragg–Bulirsch–Stoer method for ordinary differential equations.

Fully Symmetric Interpolatory Rules for Multiple Integrals

Alan Genz

SIAM J. Numer. Anal. 23, pp. 1273-1283 (11 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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A method is given for the direct determination of the weights for fully symmetric integration rules for the hypercube, using multivariable Lagrange interpolation polynomials. The formulas for the weights lead to new classes of efficient rules.

Extensions of Some Results for Interpolatory Product Integration Rules to Rules not Necessarily of Interpolatory Type

Catterina Dagnino

SIAM J. Numer. Anal. 23, pp. 1284-1289 (6 pages)

Online Publication Date: July 14, 2006

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This note is concerned with the product integration rules for the numerical evaluation of $\int_{ - 1}^1 {\omega (x)K(x)f(x)dx} $, where $\omega (x) > 0$ on $( - 1,1)$, ${K(x)}$ is Lebesgue integrable and $f$ is at least Riemann integrable (preferably smooth). We recall an interesting convergence theorem proved by Sloan and Smith and stated only for a class of interpolatory rules, and we show that it can be generalized to a larger set of interpolatory and noninterpolatory rules. This latter result is then applied to some special classes of quadratures.

On the Computation of the Coefficients of $s$-Orthogonal Polynomials

Graziano Vincenti

SIAM J. Numer. Anal. 23, pp. 1290-1294 (5 pages)

Online Publication Date: July 14, 2006

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In this paper we discuss an iterative process to compute the coefficients of $s$-orthogonal polynomials on $[ - c,c]$ with respect to an even weight function. This process is applied to Legendre $s$-polynomials; related numerical results are listed in the following cases: $n = 2,3$, $1 \leqq s \leqq 10$; $n = 4,5$, $1 \leqq s \leqq 5$; $n = 6,7$, $1 \leqq s \leqq 3$; $n = 8,9$, $1 \leqq s \leqq 2$; $n = 10,11$, $s = 1$.

The Geodesics of Pryce’s Relative Distance in $\mathbb{R}_\infty ^2 $

W. Govaerts

SIAM J. Numer. Anal. 23, pp. 1295-1302 (8 pages)

Online Publication Date: July 14, 2006

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J. D. Pryce defined a new measure of relative error for vectors. His function $\rho (x,y)$, where $x$, $y$ are two nonzero vectors in a Banach space $E$, turns $E\backslash \{ 0 \}$ into a metric space. Moreover, $\rho (x,y)$ is asymptotically equivalent to $\rho _0 (x,y) = {{\|x - y\|} / {\|x\|}}$.
We solve a question of J. D. Pryce [this Journal, 21 (1984), pp. 202–215] by describing the geodesic lines in $\mathbb{R}_\infty ^2 \backslash \{ 0 \}$ with respect to $\rho $. A path with minimal length is found between any two points in $\mathbb{R}_\infty ^2 \backslash \{ 0 \}$ which makes effective computation of distances possible. As a surprising consequence we find $\rho (x, - x) = 4\ln 2$ for all nonzero $x$.
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