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SIAM J. on Numerical Analysis

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1993

Volume 30, Issue 6, pp. 1537-1838


Domain Decomposition Type Iterative Techniques for Parabolic Problems on Locally Refined Grids

Richard E. Ewing, Raytcho D. Lazarov, Joseph E. Pasciak, and Panayot S. Vassilevski

SIAM J. Numer. Anal. 30, pp. 1537-1557 (21 pages) | Cited 7 times

Online Publication Date: July 14, 2006

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Based on an extension of the discontinuous Galerkin finite element method, discretization schemes for solving parabolic problems on grids with local refinement, both in space and in time, are constructed. The stability of schemes constructed in this way is automatically ensured by the method. The construction of two-level preconditioners utilizing local timestepping and a global coarse-grid solver both on standard, rectangular, and uniform grids, is the main objective of the paper. The optimal convergence properties of such two-level preconditioners are studied. The theory is illustrated by a set of numerical examples.

A Quasi-Monte Carlo Approach to Particle Simulation of the Heat Equation

William J. Morokoff and Russel E. Caflisch

SIAM J. Numer. Anal. 30, pp. 1558-1573 (16 pages) | Cited 14 times

Online Publication Date: July 14, 2006

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The convergence of the Monte Carlo method for numerical integration can often be improved by replacing random numbers with more uniformly distributed numbers known as quasi-random. In this paper the convergence of Monte Carlo particle simulation is studied when these quasi-random sequences are used. For the one-dimensional heat equation discretized in both space and time, convergence is proved for a quasi-random simulation using reordering of the particles according to their position. Experimental results are presented for the spatially continuous heat equation in one and two dimensions. The results indicate that a significant improvement in both magnitude of error and convergence rate can be achieved over standard Monte Carlo simulations for certain low-dimensional problems.

Estimation of Variable Cefficients in the Fokker–Planck Quations Using Moving Node Finite Elements

H. T. Banks, H. T. Tran, and D. E. Woodward

SIAM J. Numer. Anal. 30, pp. 1574-1602 (29 pages) | Cited 3 times

Online Publication Date: July 14, 2006

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Inverse problems are considered for the estimation of temporally and spatially varying coefficients in the Fokker–Planck or forward Kolmogorov equation that often arise in size/age structured population models. These are difficult problems even for simulation studies since they share certain numerical difficulties with transport-dominated diffusion-convection problems of fluid dynamics. These difficulties stem from the fact that when the convection is large compared to the diffusion, both traditional finite difference or finite element methods produce erroneous oscillatory solutions. In this paper, computational techniques are presented that combine a variation of the moving finite element method with spline approximations for the parameter estimation problems. Computational details of our numerical algorithm along with theoretical convergence results are presented. Several numerical examples illustrating the effectiveness of the method are also given.

On the Question of Turbulence Modeling by Approximate Inertial Manifolds and the Nonlinear Galerkin Method

John G. Heywood and Rolf Rannacher

SIAM J. Numer. Anal. 30, pp. 1603-1621 (19 pages) | Cited 3 times

Online Publication Date: July 14, 2006

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This paper contributes to the discussion about the ability of the AIM method to model turbulent flow, and about the theoretical potential of the AIM/NGM to provide a computational basis for the calculation of turbulent flow.

The Global Dynamics of Discrete Semilinear Parabolic Equations

C. M. Elliott and A. M. Stuart

SIAM J. Numer. Anal. 30, pp. 1622-1663 (42 pages) | Cited 26 times

Online Publication Date: July 14, 2006

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A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered. The gradient structure of the problem implies that, provided all steady states are isolated, solutions approach a steady state as $t \to \infty $. The dynamical properties of various finite difference and finite element schemes for the equations are analysed. The existence of absorbing sets, bounded independently of the mesh size, is proved for the numerical methods. Discrete Lyapunov functions are constructed to show that, under appropriate conditions on the mesh parameters, numerical orbits approach steady state solutions as discrete time increases. However, it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented. Furthermore, it is also shown that the explicit Euler scheme introduces spurious solutions with period 2 in the timestep. As a result, the absorbing set is destroyed and there is initial data leading to blow up of the scheme, however small the mesh parameters are taken. To obtain stabilization to a steady state for this scheme, it is necessary to restrict the timestep in terms of the initial data and the space step. Implicit schemes are constructed for which absorbing sets and Lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step; both one-step and multistep (BDF) methods are studied.

A New Scheme for the Approximation of Advection-Diffusion Equations by Collocation

Daniele Funaro

SIAM J. Numer. Anal. 30, pp. 1664-1676 (13 pages) | Cited 11 times

Online Publication Date: July 14, 2006

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A competitive algorithm, which allows the computation of approximated polynomial solutions of advection-diffusion equations in the square, is presented. The equation is collocated at a special grid and the corresponding system is solved by a low-cost preconditioned iterative procedure. The method provides accurate results even when the solution presents sharp boundary layers.

Numerical Methods for the Simulation of Flow in Root-Soil Systems

Todd Arbogast, Mandri Obeyesekere, and Mary F. Wheeler

SIAM J. Numer. Anal. 30, pp. 1677-1702 (26 pages) | Cited 7 times

Online Publication Date: July 14, 2006

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The numerical properties of approximation schemes for a model that simulates water transport in root-soil systems are considered. The model is derived in detail. It is based on a previously proposed model which is reformulated completely in terms of the water potential. The system of equations consists of a parabolic partial differential equation that contains a nonlinear capacity term coupled to two linear ordinary differential equations. A closed form solution is obtained for one of the latter equations. Finite element and finite difference schemes are defined to approximate the solution of the coupled system. Some new techniques which have wide applicability for analyzing the nonlinear capacity term are used, and optimal order error estimates are derived. A postprocessed water mass flux computation is also presented and shown to be superconvergent to the true flux. Computational results which verify the theoretical convergence rates are given.

Solution of Nonlinear Diffusion Problems by Linear Approximation Schemes

J. Kavčur, A. Handlovivčová, and M. Kavčurová

SIAM J. Numer. Anal. 30, pp. 1703-1722 (20 pages) | Cited 7 times

Online Publication Date: July 14, 2006

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An approximation of degenerate parabolic problems (of porous medium and Stefan type) is studied. The enthalpy formulation and the variational technique are used. Nonstandard semidiscretization in time is used and Newton-like iterations are applied to solve the corresponding elliptic problems. Some numerical experiments are discussed and compared to other methods and to analytic solutions.

Application of Global Methods in Parallel Shooting

M. E. Kramer and R. M. M. Mattheij

SIAM J. Numer. Anal. 30, pp. 1723-1739 (17 pages) | Cited 1 time

Online Publication Date: July 14, 2006

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A new divide and conquer method is proposed. It employs a coarse grid discretization like multiple shooting does, but solves the local problems in a “BVP-way,” i.e., by some global method. A sophisticated error control is developed to combine local and global convergence of the Newton updating. An implementation based on using the collocation code COLNEW is briefly discussed and a number of examples are given to illustrate the success of the method, in particular for singular perturbation problems.

FFT-Based Preconditioners for Toeplitz-Block Least Squares Problems

Raymond H. Chan, James G. Nagy, and Robert J. Plemmons

SIAM J. Numer. Anal. 30, pp. 1740-1768 (29 pages) | Cited 20 times

Online Publication Date: July 14, 2006

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Discretized two-dimensional deconvolution problems arising, e.g., in image restoration and seismic tomography, can be formulated as least squares computations, $\min \| {b - Tx} \|_2 $, where $T$ is often a large-scale rectangular Toeplitz-block matrix. The authors consider solving such block least squares problems by the preconditioned conjugate gradient algorithm using square nonsingular circulant-block and related preconditioners, constructed from the blocks of the rectangular matrix $T$. Preconditioning with such matrices allows efficient implementation using the one-dimensional or two-dimensional fast Fourier transform (FFT). Two-block preconditioners, related to those proposed by T. Chan and J. Olkin for square nonsingular Toeplitz-block systems, are derived and analyzed. It is shown that, for important classes of $T$, the singular values of the preconditioned matrix are clustered around one. This extends the authors’ earlier work on preconditioners for Toeplitz least squares iterations for one-dimensional problems.
It is well known that the resolution of ill-posed deconvolution problems can be substantially improved by regularization to compensate for their ill-posed nature. It is shown that regularization can easily be incorporated into our preconditioners, and a report is given on numerical experiments on a Cray Y-MP. The experiments illustrate good convergence properties of these FFT-based preconditioned iterations.

Quadrature Methods for Strongly Elliptic Equations of Negative Order on Smooth Closed Curves

J. Saranen and L. Schroderus

SIAM J. Numer. Anal. 30, pp. 1769-1795 (27 pages) | Cited 10 times

Online Publication Date: July 14, 2006

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In this article the authors introduce and analyze a family of simple quadrature methods for strongly elliptic equations of negative order. The results cover all the classical pseudodifferential equations of this kind, and hence the boundary integral equations appearing in applications. The quadrature methods are based on the trapezoidal approximation of the operator with respect to a uniform mesh, and on $\varepsilon $-collocation. They are proved to have the stability and the convergence with the maximal rate $O(h^{ - \beta } )$ of convergence in general, and $O(h^{ - \beta + 1} )$ for some special values of $\varepsilon $. Here $\beta $ is the order of the operator. The method is applicable to systems of boundary integral equations as well. In this connection the authors have conducted some numerical experiments, which confirm their theoretical results.

Optimal a Posteriori Parameter Choice for Tikhonov Regularization for Solving Nonlinear Ill-Posed Problems

O. Scherzer, H. W. Engl, and K. Kunisch

SIAM J. Numer. Anal. 30, pp. 1796-1838 (43 pages) | Cited 43 times

Online Publication Date: July 14, 2006

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The authors propose an a-posteriors strategy for choosing the regularization parameter in Tikhonov regularization for solving nonlinear ill-posed problems and show that under certain conditions, the convergence rate obtained with this strategy is optimal. As a by-product, a new stability estimate for the regularized solutions is given which applies to a class of parameter identification problems. The authors compare the parameter choice strategy with Morozov’s Discrepancy Principle. Finally, numerical results are presented.
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