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SIAM J. on Numerical Analysis

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2008

Volume 46, Issue 6, pp. 2769-3312


A New Approach to Numerical Solution of Defective Boundary Value Problems in Incompressible Fluid Dynamics

Luca Formaggia, Alessandro Veneziani, and Christian Vergara

SIAM J. Numer. Anal. 46, pp. 2769-2794 (26 pages) | Cited 1 time

Online Publication Date: July 03, 2008

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We consider the incompressible Navier–Stokes equations where on a part of the boundary flow rate and mean pressure boundary conditions are prescribed. There are basically two strategies for solving these defective boundary problems: the variational approach (see J. Heywood, R. Rannacher, and S. Turek, Internat. J. Numer. Methods Fluids, 22 (1996), pp. 325–352) and the augmented formulation (see L. Formaggia, J. F. Gerbeau, F. Nobile, and A. Quarteroni, SIAM J. Numer. Anal., 40 (2002), pp. 376–401, and A. Veneziani and C. Vergara, Internat. J. Numer. Methods Fluids, 47 (2005), pp. 803–816). However, both of these approaches present some drawbacks. For the flow rate problem, the former resorts to nonstandard functional spaces, which are quite difficult to discretize. On the other hand, for the mean pressure problem, it yields exact solutions only in very special cases. The latter is applicable only to the flow rate problem, since for the mean pressure problem it provides unfeasible boundary conditions. In this paper, we propose a new strategy based on a reformulation of the problems at hand in terms of the minimization of an appropriate functional. This approach allows us to treat the two kinds of problems (flow rate and mean pressure) successfully within the same framework, which can be useful in view of a mixed problem where the two conditions are simultaneously prescribed on different artificial boundaries. Moreover, it is more versatile, being prone to be extended to other kinds of defective conditions. We analyze the problems obtained with this approach and propose some algorithms for their numerical solution. Several numerical results are presented supporting the effectiveness of our approach.

Leapfrog for Optimal Control

C. Yalçin Kaya and J. Lyle Noakes

SIAM J. Numer. Anal. 46, pp. 2795-2817 (23 pages) | Cited 1 time

Online Publication Date: July 03, 2008

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The leapfrog algorithm, so called because of its geometric nature, for solving a class of optimal control problems is proposed. Initially a feasible trajectory is given and subdivided into smaller pieces. In each subdivision, with the assumption that local optimal controls can easily be calculated, a piecewise-optimal trajectory is obtained. Then the junctions of these smaller pieces of optimal control trajectories are updated through a scheme of midpoint maps. Under some broad assumptions the sequence of trajectories is shown to converge to a trajectory that satisfies the maximum principle. The main advantages of the leapfrog algorithm are that (i) it does not need an initial guess for the costates and (ii) the piecewise-optimal trajectory generated in each iteration is feasible. These are illustrated through a numerical implementation of leapfrog on a problem involving the van der Pol system.

Primal and Dual Interface Concentrated Iterative Substructuring Methods

Sven Beuchler, Tino Eibner, and Ulrich Langer

SIAM J. Numer. Anal. 46, pp. 2818-2842 (25 pages)

Online Publication Date: August 01, 2008

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This paper is devoted to the fast solution of interface concentrated finite element equations. The interface concentrated finite element schemes are constructed on the basis of a nonoverlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain. Similar to boundary element domain decomposition methods, the total number of unknowns per subdomain behaves like $O((H/h)^{(d-1)})$, where $H$, $h$, and $d$ denote the usual scaling parameter of the subdomains, the average discretization parameter of the subdomain boundaries, and the spatial dimension, respectively. We propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost the same complexity as the number of unknowns. In particular, the so-called all-floating finite element tearing and interconnecting solvers are highly parallel and very robust with respect to large coefficient jumps.

Analysis of Finite Element Domain Embedding Methods for Curved Domains using Uniform Grids

Sheng Zhang

SIAM J. Numer. Anal. 46, pp. 2843-2866 (24 pages)

Online Publication Date: August 01, 2008

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We analyze the error of a finite element domain embedding method for elliptic equations on a domain $\omega$ with curved boundary. The domain is embedded in a rectangle $R$ on which uniform mesh and linear continuous elements are employed. The numerical scheme is based on an extension of the differential equation from $\omega$ to $R$ by regularization with a small parameter $\epsilon$ (for Neumann and Robin problems), or penalty with a large parameter $\epsilon^{-1}$ (for the Dirichlet problem), or a mixture of these (for a mixed boundary value problem). For Neumann and Robin problems, we prove that when $\epsilon\le h$ (the mesh size), the error in the $H^1(\omega)$ norm is of the optimal order $\mathcal{O}(h)$. For the Dirichlet problem, when $\epsilon\le h^{1/2}$, the error is $\mathcal{O}(h^{1/2})$ that is not optimal. If the mesh is adjusted around $\partial\omega$ to fit it, then the optimal convergence rate $\mathcal{O}(h)$ holds for the Dirichlet problem if $\omega$ is convex and $\epsilon\le h$. If $\omega$ is not convex, then the convergence rate can only be improved to $\mathcal{O}(h^{2/3})$ by such mesh adjustment, with the parameter being $\epsilon=h^{2/3}$. In this latter case, a parameter smaller than $h^{2/3}$ thwarts the convergence rate, which is verified by a numerical result.

Reduced-Order Optimal Control Based on Approximate Inertial Manifolds for Nonlinear Dynamical Systems

Kazufumi Ito and Karl Kunisch

SIAM J. Numer. Anal. 46, pp. 2867-2891 (25 pages)

Online Publication Date: August 06, 2008

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A reduced-order method for optimal control problems in infinite dimensions based on approximate inertial manifolds is developed. Convergence of the cost, optimal controls, and optimal states of the finite dimensional, reduced-order, optimal control problems to the original optimal control problem is analyzed. Special attention is given to the particular case when the dynamics are described by the Navier–Stokes equations in dimension two.

Efficient Strong Integrators for Linear Stochastic Systems

Gabriel Lord, Simon J. A. Malham, and Anke Wiese

SIAM J. Numer. Anal. 46, pp. 2892-2919 (28 pages)

Online Publication Date: August 06, 2008

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We present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of Wiener processes. These schemes are based on the Neumann (stochastic Taylor) and Magnus expansions. First, we consider the case when the governing linear diffusion vector fields commute with each other, but not with the linear drift vector field. We prove that numerical methods based on the Magnus expansion are more accurate in the mean-square sense than corresponding stochastic Taylor integration schemes. Second, we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations. Consequently, for general nonlinear stochastic differential equations with noncommuting vector fields, we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order. Third, we consider the consequences in two numerical studies, one of which is an application arising in stochastic linear-quadratic optimal control.

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

E. Carlini, M. Falcone, N. Forcadel, and R. Monneau

SIAM J. Numer. Anal. 46, pp. 2920-2952 (33 pages) | Cited 2 times

Online Publication Date: August 20, 2008

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We present a new fast-marching algorithm for an eikonal equation with a velocity-changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the fast-marching method in two respects. The first is that the new scheme can deal with a time-dependent velocity, and the second is that there is no restriction on its change in sign. We analyze the properties of the algorithm, and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in $\mathbb{R}^2$.

ENO Multiresolution Schemes with General Discretizations

Pascal Getreuer and François G. Meyer

SIAM J. Numer. Anal. 46, pp. 2953-2977 (25 pages)

Online Publication Date: August 20, 2008

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Harten's framework is a nonlinear generalization of the wavelet framework. Previously, the choice of discretization (scaling function) in Harten multiresolution schemes has been limited to point-value, cell-average, and hat-based discretization. This paper shows how to construct multiresolution schemes consistent with Harten's framework for a variety of discretizations. The construction here begins with the discrete operators and deduces the corresponding continuous operators, reversing the order of the usual approach. This construction yields as a special case essentially nonoscillatory (ENO) multiresolution schemes for any order of spline discretization and also has the flexibility to define multiresolution schemes with nonspline discretizations. An error-control strategy is also developed.

Numerical Approximation of Relative Equilibria for Equivariant PDEs

Vera Thümmler

SIAM J. Numer. Anal. 46, pp. 2978-3005 (28 pages) | Cited 1 time

Online Publication Date: August 20, 2008

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We prove convergence results for the numerical approximation of relative equilibria of parabolic systems in one space dimension. These systems are special examples of equivariant evolution equations. We use finite differences on a large interval with appropriately chosen boundary conditions. Moreover, we consider the approximation of isolated eigenvalues of finite multiplicity of the linear operator which arises from linearization at the equilibrium as well as the approximation of the corresponding invariant subspace. The results in this paper which are a generalization of the results in [V. Thümmler, Numerical Analysis of the Method of Freezing Traveling Waves, Ph.D. thesis, Bielefeld University, 2005] are illustrated by numerical computations for the cubic quintic Ginzburg–Landau equation.

Polynomial Extension Operators. Part I

Leszek Demkowicz, Jayadeep Gopalakrishnan, and Joachim Schöberl

SIAM J. Numer. Anal. 46, pp. 3006-3031 (26 pages) | Cited 2 times

Online Publication Date: September 04, 2008

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In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from $H^{1/2}(\partial K)$ into $H^1(K)$, for any tetrahedron $K$.

Finite Volume Method for 2D Linear and Nonlinear Elliptic Problems with Discontinuities

Franck Boyer and Florence Hubert

SIAM J. Numer. Anal. 46, pp. 3032-3070 (39 pages) | Cited 3 times

Online Publication Date: September 04, 2008

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In this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the discrete duality finite volume (DDFV) framework. This family of schemes allows very general meshes and inherits the main properties of the continuous problem. In order to take into account the discontinuities and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first illustrate our approach by the study of the 1D situation. Then, we show how to design our new scheme, called m-DDFV, and we propose its analysis. We also describe an iterative solver, whose convergence is proved, which can be used to solve the nonlinear discrete equations defining the finite volume scheme. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for both linear and nonlinear problems.

Projected Pseudotransient Continuation

C. T. Kelley, Li-Zhi Liao, Liqun Qi, Moody T. Chu, J. P. Reese, and C. Winton

SIAM J. Numer. Anal. 46, pp. 3071-3083 (13 pages) | Cited 1 time

Online Publication Date: September 04, 2008

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We propose and analyze a pseudotransient continuation algorithm for dynamics on subsets of $R^N$. Examples include certain flows on manifolds and the dynamic formulation of bound-constrained optimization problems. The method gets its global convergence properties from the dynamics and inherits its local convergence properties from any fast locally convergent iteration.

Convergence of Fourier-Wavelet Models for Gaussian Random Processes

O. Kurbanmuradov and K. Sabelfeld

SIAM J. Numer. Anal. 46, pp. 3084-3112 (29 pages)

Online Publication Date: September 04, 2008

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Mean square convergence and convergence in probability of Fourier-wavelet models (FWM) of stationary Gaussian random processes in the metric of Banach space of continuously differentiable functions and in Sobolev space are studied. Sufficient conditions for the convergence formulated in the frame of spectral functions are given. It is shown that the given rates of convergence of FWM in the mean square obtained in the Nikolskiĭ–Besov classes cannot be improved.

Operator Splitting for an Immunology Model Using Reaction-Diffusion Equations with Stochastic Source Terms

Timothy A. Lucas

SIAM J. Numer. Anal. 46, pp. 3113-3135 (23 pages) | Cited 1 time

Online Publication Date: September 04, 2008

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When immune cells detect foreign molecules, they secrete soluble factors that attract other immune cells to the site of the infection. In this paper, I study numerical solutions to a model of this behavior proposed by Kepler. In this model the soluble factors are governed by a system of reaction-diffusion equations with sources that are centered on the cells. The motion of the model cells is a Langevin process that is biased toward the gradient of the soluble factors. I have shown that the solution to this system exists for all time and remains positive, the supremum is a priori bounded, and the derivatives are bounded for finite time. I have also developed a first order split scheme for solving the reaction-diffusion stochastic system. This allows us to make use of known first order schemes for solving the diffusion, the reaction, and the stochastic differential equations separately.

An A Posteriori Condition on the Numerical Approximations of the Navier–Stokes Equations for the Existence of a Strong Solution

Masoumeh Dashti and James C. Robinson

SIAM J. Numer. Anal. 46, pp. 3136-3150 (15 pages)

Online Publication Date: September 04, 2008

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In their 2006 paper, Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. prove that a sufficiently smooth strong solution of the 3D Navier–Stokes equations is robust with respect to small enough changes in initial conditions and forcing function. They also show that if a regular enough strong solution exists, then Galerkin approximations converge to it. They then use these results to conclude that the existence of a sufficiently regular strong solution can be verified using sufficiently refined numerical computations. In this paper we study the strong solutions with less regularity than those considered in Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. We prove a similar robustness result and show the validity of the results relating convergent numerical computations and the existence of the strong solutions.

Taming the CFL Number for Discontinuous Galerkin Methods on Structured Meshes

T. Warburton and T. Hagstrom

SIAM J. Numer. Anal. 46, pp. 3151-3180 (30 pages)

Online Publication Date: September 17, 2008

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The upwind discontinuous Galerkin method is an attractive method for solving time-dependent hyperbolic conservation laws. It is possible to use high-order explicit time-stepping methods and high-order spatial approximations without incurring heavy numerical linear algebra overheads. However, the Courant–Friedrichs–Lewy (CFL) condition for these methods depends on the polynomial order used, and there is a somewhat excessive cost for using very high order spatial approximation. We discuss the impact of a covolume mesh based filter on the CFL number for these methods and present an algorithm which has a CFL number independent of the spatial order of approximation. We present computational results for the advection equation and the wave equation on one-dimensional meshes using up to tenth order in space and time.

The Composite Mini Element—Coarse Mesh Computation of Stokes Flows on Complicated Domains

Daniel Peterseim and Stefan A. Sauter

SIAM J. Numer. Anal. 46, pp. 3181-3206 (26 pages)

Online Publication Date: September 17, 2008

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We introduce a new finite element method, the composite mini element, for the mixed discretization of the Stokes equations on two- and three-dimensional domains that may contain a huge number of geometric details. In standard finite element discretizations of the Stokes problem, such as the classical mini element, the approximation quality is determined by the maximal mesh size of the underlying triangulation, while the computational effort is determined by its number of elements. If the physical domain is very complicated, then the minimal number of simplices, which are necessary to resolve the domain, can be very large and distributed in a nonoptimal way with respect to the approximation quality. In contrast to that, the minimal dimension of the composite mini element space is independent of the number of geometric details. Instead of a geometric resolution of the domain and the boundary condition by the finite element mesh the shape of the finite element functions is adapted to the geometric details. This approach allows low-dimensional approximations even for problems with complicated geometric details such as holes or rough boundaries. We prove its linear (optimal order) approximability and its inf-sup stability. Further, we will be able to control the nonconformity in the space without increasing the space dimension in such a way that the a priori error estimate $\|{\mathbf{u}-\mathbf{u}^{\mathrm{CME}}}\|_{1,\Omega}+\|{p-p^{\mathrm{CME}}} \|_{0,\Omega}\lesssim h\|{\mathbf{f}}\|_{0,\Omega}$ holds. Thereby, in contrast to the classical methods, the choice of the mesh size parameter $h$ is not constrained by the size of geometric details. In addition, it turns out that the method can be viewed as a coarse-scale generalization of the classical mini element approach; i.e., it reduces the computational effort, while the approximation quality depends on the (coarse) mesh size in the usual way.

Fully Computable Bounds for the Error in Nonconforming Finite Element Approximations of Arbitrary Order on Triangular Elements

Mark Ainsworth and Richard Rankin

SIAM J. Numer. Anal. 46, pp. 3207-3232 (26 pages) | Cited 1 time

Online Publication Date: October 17, 2008

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We obtain a fully computable a posteriori error bound on the broken energy norm of the error in the nonconforming finite element approximation on triangles of arbitrary order of a linear second order elliptic problem with variable permeability. The estimator is completely free of unknown constants and provides a guaranteed numerical bound on the broken energy norm of the error. This estimator is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms.

Convergence of a Semidiscrete Two-Grid Algorithm for the Controllability of the $1-d$ Wave Equation

Mihaela Negreanu

SIAM J. Numer. Anal. 46, pp. 3233-3263 (31 pages)

Online Publication Date: October 17, 2008

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The problem of exact controllability of elastic strings has been extensively studied during the last years. We consider the problem of computing numerically the boundary control for a finite-dimensional system obtained by discretizing in space the $1-d$ wave equation. More precisely, we analyze whether the controls of numerical approximation schemes converge to the control of the continuous wave equation as the mesh size tends to zero. It is by now well known that, due to high-frequency spurious oscillations, numerical instabilities occur and may lead to the failure of convergence of some apparently natural numerical algorithms. In other words, the classical convergence property of numerical schemes does not guarantee a stable and convergent approximation of controls. Several remedies have been proposed in the literature to compensate for this fact: Tychonoff regularization, Fourier filtering, and mixed finite elements. In this paper we prove that the two-grid method proposed by Glowinski in [J. Comput. Phys., 103 (1992), pp. 189–221] to numerically approximate the control of the wave equation converges in $1-d$. We prove this result in the context of the finite-element space semidiscretization. Our method of proof relies essentially on the particular properties of the Fourier representation of the initial data of the coarse mesh when projected into the fine one. The explicit representation formula of the solutions shows that the high-frequency components are modulated by some weights that diminish the effect of these spurious components. This fact, combined with discrete multipliers techniques, allows us to prove uniform observability inequalities. Classical arguments then allow proving the uniform boundedness of the controls and passing to the limit as the mesh size tends to zero. In this way we prove the convergence of the controls of the finite-element semidiscrete approximation of the $1-d$ wave equation with a boundary control on one extreme. It is important to underline that the controls obtained by this two-grid algorithm are not exact in the sense that they only guarantee the controllability of the projection of the solutions into the coarse mesh.

Solving Elliptic Finite Element Systems in Near-Linear Time with Support Preconditioners

Erik G. Boman, Bruce Hendrickson, and Stephen Vavasis

SIAM J. Numer. Anal. 46, pp. 3264-3284 (21 pages) | Cited 1 time

Online Publication Date: October 17, 2008

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We consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems. Our main result is that these linear systems, which are symmetric and positive semidefinite, are well approximated by symmetric diagonally dominant matrices. Our framework for defining matrix approximation is support theory. Significant graph theoretic work has already been developed in the support framework for preconditioners in the diagonally dominant case, and, in particular, it is known that such systems can be solved with iterative methods in nearly linear time. Thus, our approximation result implies that these graph theoretic techniques can also solve a class of finite element problems in nearly linear time. We show that the support number bounds, which control the number of iterations in the preconditioned iterative solver, depend on mesh quality measures but not on the problem size or shape of the domain.

Effective Laguerre Asymptotics

David Borwein, Jonathan M. Borwein, and Richard E. Crandall

SIAM J. Numer. Anal. 46, pp. 3285-3312 (28 pages)

Online Publication Date: October 17, 2008

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It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs $(a,z)$ one has the large-$n$ asymptotic behavior $L_n^{(-a)}(-z) \sim C(a,z) n^{-a/2-1/4} e^{2\sqrt{nz}}$. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an “exp-arc” method giving a natural bridge between converging series and effective asymptotics. (A corrected version of this paper has been appended to the originally posted pdf).
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