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SIAM J. on Numerical Analysis

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2011

Volume 49, Issue 3, pp. 917-1319


Error Analysis of a Fractional Time-Stepping Technique for Incompressible Flows with Variable Density

J.-L. Guermond and Abner J. Salgado

SIAM J. Numer. Anal. 49, pp. 917-944 (28 pages)

Online Publication Date: May 10, 2011

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In this paper we analyze the convergence properties of a new fractional time-stepping technique for the solution of the variable density incompressible Navier–Stokes equations. The main feature of this method is that, contrary to other existing algorithms, the pressure is determined by just solving one Poisson equation per time step. First-order error estimates are proved, and stability of a formally second-order variant of the method is established.

An Energy Stable and Convergent Finite-Difference Scheme for the Modified Phase Field Crystal Equation

C. Wang and S. M. Wise

SIAM J. Numer. Anal. 49, pp. 945-969 (25 pages)

Online Publication Date: May 12, 2011

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We present an unconditionally energy stable finite difference scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The method is based on a convex splitting of a discrete pseudoenergy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step-size. We present a local-in-time error estimate that ensures the pointwise convergence of the scheme.

Quasi-Optimality of Adaptive Nonconforming Finite Element Methods for the Stokes Equations

Roland Becker and Shipeng Mao

SIAM J. Numer. Anal. 49, pp. 970-991 (22 pages)

Online Publication Date: May 12, 2011

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We prove convergence and quasi-optimal complexity of adaptive nonconforming low-order finite element methods for the Stokes equations, covering the Crouzeix–Raviart discretization on triangular and tetrahedral meshes, as well as the Rannacher–Turek discretization on two- and three-dimensional rectangular meshes. Hanging nodes are allowed in order to ease local mesh refinement. The adaptive algorithm is based on standard a posteriori error estimators consisting of two parts: a volume residual and an edge term measuring the nonconformity of the velocity approximation. We use an adaptive marking strategies, which, in each step of the iteration, takes only the dominant term into account. This paper can be regarded as an extension of [R. Becker, S. Mao, and Z.-C. Shi, SIAM J. Numer. Anal., 47 (2010), pp. 4639–4659] to the Stokes problem, but the analysis here does not make use of any relationship between mixed and nonconforming finite element methods.

A Priori Mesh Grading for an Elliptic Problem with Dirac Right-Hand Side

Thomas Apel, Olaf Benedix, Dieter Sirch, and Boris Vexler

SIAM J. Numer. Anal. 49, pp. 992-1005 (14 pages)

Online Publication Date: May 19, 2011

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The Green function of the Poisson equation in two dimensions is not contained in the Sobolev space $H^1(\Omega)$ such that finite element error estimates for the discretization of a problem with the Dirac measure on the right hand-side are nonstandard and quasi-uniform meshes are inappropriate. By using graded meshes $L^2$-error estimates of almost optimal order are shown. As a byproduct, we show for the Poisson equation with a right-hand side in $L^2$ that appropriate mesh refinement near some interior point diminishes the error at this point by nearly one order.

Analysis of a Modified First-Order System Least Squares Method for Linear Elasticity with Improved Momentum Balance

Gerhard Starke, Alexander Schwarz, and Jörg Schröder

SIAM J. Numer. Anal. 49, pp. 1006-1022 (17 pages)

Online Publication Date: May 19, 2011

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A modified first-order system least squares formulation for linear elasticity, obtained by adding the antisymmetric displacement gradient in the test space, is analyzed. This approach leads to surprisingly small momentum balance error compared to standard least squares approaches. It is shown that the modified least squares formulation is well posed and its performance is illustrated by adaptive finite element computation based on using a closely related least squares functional as a posteriori error estimator. The results of our numerical computations show that, for the modified least squares approach, the momentum balance error converges at a much faster rate than the overall error. We prove that this is due to a strong connection of the stress approximation to that obtained from a mixed formulation based on the Hellinger–Reissner principle (with exact local momentum balance). The practical significance is that our proposed approach is almost momentum-conservative at a smaller number of degrees of freedom than mixed approximations with exact local momentum balance.

Constraint Preserving Schemes Using Potential-Based Fluxes. II. Genuinely Multidimensional Systems of Conservation Laws

Siddhartha Mishra and Eitan Tadmor

SIAM J. Numer. Anal. 49, pp. 1023-1045 (23 pages)

Online Publication Date: May 24, 2011

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We introduce a class of numerical schemes that preserve a discrete version of vorticity in conservation laws which involve grad advection. These schemes are based on reformulating finite volume schemes in terms of vertex centered numerical potentials. The resulting potential-based schemes have a genuinely multidimensional structure. A suitable choice of potentials leads to discrete vorticity preserving schemes that are simple to code, computationally inexpensive, and proven to be stable. We extend our discussion to other classes of genuinely multidimensional schemes. Numerical examples for linear grad advection equations, linear and nonlinear wave equation systems, and the Euler equations of gas dynamics are presented.

Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues

A. H. Barnett and A. Hassell

SIAM J. Numer. Anal. 49, pp. 1046-1063 (18 pages)

Online Publication Date: May 24, 2011

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We study eigenfunctions $\phi_j$ and eigenvalues $E_j$ of the Dirichlet Laplacian on a bounded domain $\Omega\subset\mathbb{R}^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E>0$ and the spectrum $\{E_j\}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(\Delta+E)u=0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler–Payne by a factor of the wavenumber $\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes (Theorem 1.3), of interest in its own right. Namely, the operator norm of the sum of rank 1 operators $\partial_n\phi_j\langle\partial_n\phi_j,\cdot\rangle$ over all $E_j$ in a spectral window of width $\sqrt{E}$—a sum with about $E^{(n-1)/2}$ terms—is at most a constant factor (independent of $E$) larger than the operator norm of any one individual term.

A Parallel Robin–Robin Domain Decomposition Method for the Stokes–Darcy System

Wenbin Chen, Max Gunzburger, Fei Hua, and Xiaoming Wang

SIAM J. Numer. Anal. 49, pp. 1064-1084 (21 pages)

Online Publication Date: May 31, 2011

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We propose a new parallel Robin–Robin domain decomposition method for the coupled Stokes–Darcy system with Beavers–Joseph–Saffman–Jones interface boundary condition. In particular, we prove that, with an appropriate choice of parameters, the scheme converges geometrically independent of the mesh size.

Convergence of a Discontinuous Galerkin Method for the Miscible Displacement Equation under Low Regularity

Beatrice M. Rivière and Noel J. Walkington

SIAM J. Numer. Anal. 49, pp. 1085-1110 (26 pages)

Online Publication Date: June 09, 2011

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Discontinuous Galerkin time discretizations are combined with the mixed finite element and continuous finite element methods to solve the miscible displacement problem. Stable schemes of arbitrary order in space and time are obtained. Under low regularity assumptions on the data, convergence of the scheme is proved by using compactness results for functions that may be discontinuous in time.

Optimizing Gaussian Quadrature for Positive Definite Strong Moment Functionals

Brian A. Hagler

SIAM J. Numer. Anal. 49, pp. 1111-1126 (16 pages)

Online Publication Date: June 09, 2011

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Classical Gaussian quadrature is anchored in the space of real polynomials at polynomial degree 0 and is monotonically directed by successively increasing polynomial degree. The results of recent research, investigating weighing anchor and charting various courses in search of the best currents in the space of real Laurent polynomials, are presented. Standard error bound minimization anchoring and steering algorithms which utilize the moments of positive definite strong moment functionals to guide the construction of ordered orthogonal Laurent polynomial sequences are introduced.

Stability of the Nyström Method for the Sherman–Lauricella Equation

Victor D. Didenko and Johan Helsing

SIAM J. Numer. Anal. 49, pp. 1127-1148 (22 pages)

Online Publication Date: June 09, 2011

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The stability of the Nyström method for the Sherman–Lauricella equation on piecewise smooth closed simple contour $\Gamma$ is studied. It is shown that in the space $L_2$ the method is stable if and only if certain operators associated with the corner points of $\Gamma$ are invertible. If $\Gamma$ does not have corner points, the method is always stable. Numerical experiments show the transformation of solutions when the unit circle is continuously transformed into the unit square, and then into various rhombuses. Examples also show an excellent convergence of the method.

Error Analysis of the SUPG Finite Element Discretization of Evolutionary Convection-Diffusion-Reaction Equations

Volker John and Julia Novo

SIAM J. Numer. Anal. 49, pp. 1149-1176 (28 pages)

Online Publication Date: June 09, 2011

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Conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the streamline-upwind Petrov–Galerkin (SUPG) finite element stabilization of time-dependent convection-diffusion-reaction equations. Exemplarily, it is shown for the SUPG method combined with the backward Euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step. The stabilization vanishes in the time-continuous limit. However, based on numerical experience, this seems not to be the correct behavior. For this reason, the main focus of the paper consists in deriving estimates in which the stabilization parameters do not depend on the length of the time step. It is shown that such estimates can be obtained in the case of time-independent convection and reaction. An error estimate for the time-continuous case with the standard order of convergence is derived for stabilization parameters of the same form as they are optimal for the steady-state problem. Analogous estimates are obtained for the fully discrete case using the backward Euler method and the Crank–Nicolson scheme. Numerical studies support the analytical results.

A Direct Algorithm for Ultrasound Imaging of Internal Corrosion

Habib Ammari, Hyeonbae Kang, Eunjoo Kim, Mikyoung Lim, and Kaouthar Louati

SIAM J. Numer. Anal. 49, pp. 1177-1193 (17 pages)

Online Publication Date: June 09, 2011

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We develop a direct (noniterative) algorithm to address the inverse problem of identifying a collection of disjoint internal corrosive parts of small Hausdorff measures in pipelines from exterior ultrasound boundary measurements. The method is based on an asymptotic expansion of the effect of the corrosion in terms of the size of the corrosive parts. We numerically test the validity of the asymptotic formula at high frequencies. We also propose a simple procedure to remove high-frequency instabilities in our inversion procedure. We illustrate our main findings with a variety of computational examples.

Differential Equations for Roaming Pseudospectra: Paths to Extremal Points and Boundary Tracking

Nicola Guglielmi and Christian Lubich

SIAM J. Numer. Anal. 49, pp. 1194-1209 (16 pages) | Cited 1 time

Online Publication Date: June 14, 2011

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When studying the $\varepsilon$-pseudospectrum of a matrix, one is often interested in computing the extremal points having maximum real part or modulus. This is a crucial step, for example, when computing the distance to instability of a stable system. Using the property that the pseudospectrum is determined via perturbations by rank-1 matrices, we derive differential equations on the manifold of normalized rank-1 matrices whose solutions tend to the critical rank-1 perturbations associated with the extremal points of (locally) maximum real part and modulus. This approach also allows us to track the boundary contour of the pseudospectrum in a neighborhood of the extremal points. The technique we propose is related to an idea recently developed by Guglielmi and Overton, who derived discrete dynamical systems instead of the continuous ones we present. The method turns out to be fast in comparison with those previously proposed in the literature and appears to be promising in dealing with large sparse problems.

Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation

J. M. Melenk and S. Sauter

SIAM J. Numer. Anal. 49, pp. 1210-1243 (34 pages) | Cited 1 time

Online Publication Date: June 23, 2011

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We develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems (bvps) by considering the Helmholtz equation at high wavenumber $k$ as our model problem. The key element in this theory is a novel $k$-explicit regularity theory for Helmholtz bvps that is based on decomposing the solution into two parts: the first part has the Sobolev regularity properties expected of second order elliptic PDEs but features $k$-independent regularity constants; the second part is an analytic function for which $k$-explicit bounds for all derivatives are given. This decomposition is worked out in detail for several types of bvps, namely, the Helmholtz equation in bounded smooth domains or convex polygonal domains with Robin boundary conditions and in exterior domains with Dirichlet boundary conditions. We present an error analysis for the classical $hp$-version of the finite element method ($hp$-FEM) where the dependence on the mesh width $h$, the approximation order $p$, and the wavenumber $k$ is given explicitly. In particular, under the assumption that the solution operator for Helmholtz problems is polynomially bounded in $k$, it is shown that quasi optimality is obtained under the conditions that $kh/p$ is sufficiently small and the polynomial degree $p$ is at least O(log $k$).

Rank-Deficient Nonlinear Least Squares Problems and Subset Selection

I. C. F. Ipsen, C. T. Kelley, and S. R. Pope

SIAM J. Numer. Anal. 49, pp. 1244-1266 (23 pages)

Online Publication Date: June 23, 2011

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We examine the local convergence of the Levenberg–Marquardt method for the solution of nonlinear least squares problems that are rank-deficient and have nonzero residual. We show that replacing the Jacobian by a truncated singular value decomposition can be numerically unstable. We recommend instead the use of subset selection. We corroborate our recommendations by perturbation analyses and numerical experiments.

Positivity and Conservation Properties of Some Integration Schemes for Mass Action Kinetics

L. Formaggia and A. Scotti

SIAM J. Numer. Anal. 49, pp. 1267-1288 (22 pages)

Online Publication Date: June 23, 2011

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The numerical schemes approximating chemical reactions according to the mass action law should reproduce at least two properties of the corresponding physical system: mass conservation and nonnegativity of the concentrations. This paper analyzes the equations of mass action kinetics providing a proof of the existence, uniqueness, and positivity of the solution under mild hypotheses on the reaction rate and the stoichiometric coefficients. We then consider some classic integration schemes in terms of conservation, positivity, and accuracy compared to schemes tailored for production-destruction systems, and propose an original scheme which guarantees conservation and nonnegativity of the solution and has order of convergence between 2 and 3.

Optimized Additive Schwarz with Harmonic Extension as a Discretization of the Continuous Parallel Schwarz Method

Felix Kwok

SIAM J. Numer. Anal. 49, pp. 1289-1316 (28 pages)

Online Publication Date: June 23, 2011

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The additive Schwarz method with harmonic extension (ASH) was introduced by Cai and Sarkis (1999) as an efficient variant of the additive Schwarz method that converges faster and requires less communication. We show that ASH can also be used with optimized transmission conditions to obtain faster convergence. We show that when the decomposition into subdomains contains no cross points, optimized ASH can be reformulated as an iteration that is closely related to the optimized Schwarz method at the continuous level. In fact, the iterates of ASH are identical to the iterates of the discretized parallel Schwarz method outside the overlap, whereas inside the overlap they are linear combinations of previous Schwarz iterates. Thus, one method converges if and only if the other one does, and they do so at the same asymptotic rate, unlike additive Schwarz, which fails to converge inside the overlap. However, when cross points are present, then ASH and the Schwarz methods are incomparable, i.e., there are cases where one method converges and the other diverges, and vice versa.

Remarks on “On a General Class of Multipoint Root-Finding Methods of High Computational Efficiency”

Miodrag S. Petković

SIAM J. Numer. Anal. 49, pp. 1317-1319 (3 pages)

Online Publication Date: June 23, 2011

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An oversight on the number of function evaluations of the $n$-point root-finding methods proposed in [SIAM J. Numer. Anal., 47 (2010), pp. 4402–4414] is discussed. To correct this flaw, modified methods of optimal computational efficiency are presented by applying higher-order Hermite interpolatory polynomials.
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