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SIAM J. on Numerical Analysis

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2002

Volume 39, Issue 6, pp. 1835-2199


Lagrangian Quadrature Schemes for Computing Weak Solutions of Quantum Stochastic Differential Equations

E. O. Ayoola

SIAM J. Numer. Anal. 39, pp. 1835-1864 (30 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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Lagrangian quadrature schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations are introduced and studied. This is accomplished within the framework of the Hudson--Parthasarathy formulation of quantum stochastic calculus and subject to matrix elements of solution being sufficiently differentiable. Results concerning convergence of these schemes in the topology of the locally convex space of solution are presented. Numerical examples are given.

On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials

Richard E. Ewing, Tao Lin, and Yanping Lin

SIAM J. Numer. Anal. 39, pp. 1865-1888 (24 pages) | Cited 52 times

Online Publication Date: July 26, 2006

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We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.

A Predicted Sequential Regularization Method for Index-2 Hessenberg DAEs

Ping Lin and Raymond J. Spiteri

SIAM J. Numer. Anal. 39, pp. 1889-1913 (25 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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The sequential regularization method (SRM) is a dynamic iterative method for the numerical solution of higher-index differential-algebraic equations (DAEs). The SRM has the advantage of being based on a regularized problem that is less stiff than those produced by standard regularization methods. Consequently, nonstiff integrators may be used, making the SRM a competitive alternative to popular integrators. In past work, the number of SRM iterations was taken to be roughly equal to the order of the numerical method used in each dynamic iteration. In this paper, we propose a predicted SRM (PSRM) that reduces the number of iterations in each dynamic iteration to one. We give a new error analysis for explicit Runge--Kutta methods applied to linear index-2 Hessenberg DAEs with or without singularities. We also give numerical examples to confirm the predicted convergence rates. For the PSRM, extrapolation formulas and methods based on the differential part of the DAEs serve as a predictor, and the SRM iteration serves as a corrector. Implementation of higher-order schemes for the PSRM makes use of continuous extensions of Runge--Kutta methods. In particular, we give a prediction scheme for the algebraic variable at intermediate stage points that suppresses order reduction in the differential variable near a singularity. Moreover, the SRM/PSRM provides new insight into operator splitting and fast convergence rates for waveform relaxation.

A Collocation Method for the Gurtin--MacCamy Equation with Finite Life-Span

Mi-Young Kim and Yonghoon Kwon

SIAM J. Numer. Anal. 39, pp. 1914-1937 (24 pages) | Cited 9 times

Online Publication Date: July 26, 2006

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A collocation method along the characteristics for a stiff problem arising from population dynamics is proposed and analyzed. It is a fourth order implicit Runge--Kutta method of two stage to the integration of the ODE along the characteristics, whose collocation points are zeros of the linearly transformed Legendre monic polynomial. Nonnegativity of the numerical solutions is shown. The stability of the method is discussed. It is proven that the scheme is convergent at a fourth order rate in the maximum norm. Several numerical examples are presented.

Suboptimal and Optimal Convergence in Mixed Finite Element Methods

Alan Demlow

SIAM J. Numer. Anal. 39, pp. 1938-1953 (16 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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An elliptic partial differential equation may be formulated in different but equivalent ways, and the mixed finite element methods derived from these formulations have different properties. We give general error estimates for two such methods, which are always optimal for the Raviart--Thomas elements, but which are suboptimal for the Brezzi--Douglas--Marini elements in one of the methods. Computational experiments show that this suboptimal estimate is sharp.

Uniform Error Analysis for Lagrange--Galerkin Approximations of Convection-Dominated Problems

Markus Bause and Peter Knabner

SIAM J. Numer. Anal. 39, pp. 1954-1984 (31 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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In this paper we present a rigorous error analysis for the Lagrange--Galerkin method applied to convection-dominated diffusion problems. We prove new error estimates in which the constants depend on norms of the data and not of the solution and do not tend to infinity in the hyperbolic limit. This is in contrast to other results in this field. For the time discretization, uniform convergence with respect to the diffusion parameter of order O(k//tn) is shown for initial values in L2 and O(k) for initial values in H2. For the spatial discretization with linear finite elements, we verify uniform convergence of order O(h2+min{h,h2/k) for data in H2. By interpolation of Banach spaces, suboptimal convergence rates are derived under less restrictive assumptions. The analysis is heavily based on a priori estimates, uniform in the diffusion parameter, for the solution of the continuous and the semidiscrete problem. They are derived in a Lagrangian framework by transforming the Eulerian coordinates completely into subcharacteristic coordinates. Finally, we illustrate the error estimates by some numerical results.

Wavelet Least Squares Methods for Boundary Value Problems

Wolfgang Dahmen, Angela Kunoth, and Reinhold Schneider

SIAM J. Numer. Anal. 39, pp. 1985-2013 (29 pages) | Cited 5 times

Online Publication Date: July 26, 2006

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This paper is concerned with least squares methods for the numerical solution of operator equations. Our primary focus is the discussion of the following conceptual issues: the selection of appropriate least squares functionals, their numerical evaluation in the special light of recent developments of wavelet methods, and a natural way of preconditioning the resulting systems of linear equations.
We describe first a general format of variational problems that are well-posed in a certain natural topology. In order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems, their formulation as a first order system, transmission problems, the system of Stokes equations, or more general saddle point problems. Particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions. We propose a unified treatment based on wavelet expansions. In particular, we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context. This provides access to "difficult" norms, efficient preconditioners and, in the case of first order systems, optimal L2 error estimates.

A Fully Discrete Approximation for Control Problems Governed by Parabolic Variational Inequalities

Maïtine Bergounioux and Housnaa Zidani

SIAM J. Numer. Anal. 39, pp. 2014-2033 (20 pages)

Online Publication Date: July 26, 2006

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In this work we consider a numerical approximation of an optimal control problem governed by variational inequalities. We use a total discretization scheme: implicit Euler discretization with respect to the time variable and finite element method for the space variable, and we give convergence results.

Residual-Based A Posteriori Error Estimate for a Nonconforming Reissner--Mindlin Plate Finite Element

Carsten Carstensen

SIAM J. Numer. Anal. 39, pp. 2034-2044 (11 pages) | Cited 8 times

Online Publication Date: July 26, 2006

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Reliable and efficient residual-based a posteriori error estimates are established for the nonconforming finite element method for the Reissner--Mindlin plate due to Arnold and Falk [SIAM J. Numer. Anal., 26 (1989), pp. 1276--1290]. The error is estimated by a computable error estimator from above and below up to multiplicative constants that depend neither on the mesh-size nor on the plate's thickness. The error is controlled in norms that are known to converge to zero in a quasi-optimal way.

Discrete-Time Orthogonal Spline Collocation Methods for Vibration Problems

Bingkun Li, Graeme Fairweather, and Bernard Bialecki

SIAM J. Numer. Anal. 39, pp. 2045-2065 (21 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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Discrete-time orthogonal spline collocation schemes are formulated and analyzed for vibration problems involving various boundary conditions. Each problem is written as a Schrödinger-type system, which is then approximated by Crank--Nicolson and/or alternating direction implicit orthogonal spline collocation schemes. These schemes are shown to be second-order accurate in time and of optimal order accuracy in space in the Hm-norm, m=1,2.

Symplectic Integration of Hamiltonian Systems with Additive Noise

G. N. Milstein, Yu. M. Repin, and M. V. Tretyakov

SIAM J. Numer. Anal. 39, pp. 2066-2088 (23 pages) | Cited 5 times

Online Publication Date: July 26, 2006

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Hamiltonian systems with additive noise possess the property of preserving symplectic structure. Numerical methods with the same property are constructed for such systems. Special attention is paid to systems with separable Hamiltonians and to second-order differential equations with additive noise. Some numerical tests are presented.

A Nondiffusive Finite Volume Scheme for the Three-Dimensional Maxwell's Equations on Unstructured Meshes

Serge Piperno, Malika Remaki, and Loula Fezoui

SIAM J. Numer. Anal. 39, pp. 2089-2108 (20 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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We prove a sufficient CFL-like condition for the L2-stability of the second-order accurate finite volume scheme proposed by Remaki for the time-domain solution of Maxwell's equations in heterogeneous media with metallic and absorbing boundary conditions. We yield a very general sufficient condition valid for any finite volume partition in two and three space dimensions. Numerical tests show the potential of this original finite volume scheme in one, two, and three space dimensions for the numerical solution of Maxwell's equations in the time-domain.

A New Family of Mixed Finite Elements for the Linear Elastodynamic Problem

E. Bécache, P. Joly, and C. Tsogka

SIAM J. Numer. Anal. 39, pp. 2109-2132 (24 pages) | Cited 7 times

Online Publication Date: July 26, 2006

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We construct and analyze a new family of quadrangular (in two dimensions) or cubic (in three dimensions) mixed finite elements for the approximation of elastic wave equations. Our elements lead to explicit schemes (via mass lumping), after time discretization, including in the case of anisotropic media. Error estimates are given for these new elements.

Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems

Paul Houston, Christoph Schwab, and Endre Süli

SIAM J. Numer. Anal. 39, pp. 2133-2163 (31 pages) | Cited 49 times

Online Publication Date: July 26, 2006

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We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, advection-reaction equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hp-optimal error bound is derived; here, we consider only advection-reaction problems which satisfy a certain (standard) positivity condition. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by $\frac{1}{2}$ a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For elementwise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments.

Square-Conservative Schemes for a Class of Evolution Equations Using Lie-Group Methods

Jing-Bo Chen, Hans Munthe-Kaas, and Meng-Zhao Qin

SIAM J. Numer. Anal. 39, pp. 2164-2178 (15 pages)

Online Publication Date: July 26, 2006

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A new method for constructing square-conservative schemes for a class of evolution equations using Lie-group methods is presented. The basic idea is as follows. First, we discretize the space variable appropriately so that the resulting semidiscrete system of equations can be cast into a system of ordinary differential equations evolving on a sphere. Second, we apply Lie-group methods to the semidiscrete system, and then square-conservative schemes can be constructed since the obtained numerical solution evolves on the same sphere. Both exponential and Cayley coordinates are used. Numerical experiments are also reported.

Filtering the Feynman--KAC Formula

Bennett L. Fox

SIAM J. Numer. Anal. 39, pp. 2179-2199 (21 pages)

Online Publication Date: July 26, 2006

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Certain linear partial differential equations involving a Laplacian operator have formulas for their pointwise solutions as functions of Brownian motion. The Feynman--Kac formula, one such, for a heat equation with an additive linear term is smoothed by computing a sequence of conditional expectations adapted to a filtration. Certain additional smoothing of one-dimensional path integrals makes sophisticated numerical rules such as Hermite and Simpson, applied in tandem, effective. The work to get the bias below an error tolerance $\xi$ is $O( \xi^{ - 1/2 \,-\, \epsilon } ) $ with $\epsilon$ an arbitrarily small positive number, whereas it is at least order $\xi^{ - 2} $ for a benchmark method that has been used in practice. By combining the algorithm that yields that bias complexity with randomized quasi-Monte Carlo, the work to get the pointwise root mean square error (RMSE) below $\xi$ is $\, O(\xi^{-7/6 \,-\, \epsilon }) $ for each fixed spatial dimension d. A slightly more intricate algorithm reduces bias complexity---leading to $O(\xi^{-20/21 \,-\, \epsilon}) $ RMSE complexity. The latter is lower than an order $\xi^{-(d+1)/2}$ sharp lower bound on complexity for conventional solvers for each fixed d, the difference increasing rapidly with d---assuming that the number of points where a solution is desired is O(1) for each d.
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