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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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On the Finite Element Approximation of $p$-Stokes Systems SIAM J. Numer. Anal. 50, pp. 373-397 (25 pages) Online Publication Date: March 08, 2012
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In this paper we study the finite element approximation of systems of $p$-Stokes type for $p \in (1,\infty)$. We derive (in some cases optimal) error estimates for finite element approximation of the velocity and for the pressure in a suitable functional setting. The results are supported by numerical experiments. |
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$\ell_1$ Minimization with Noisy Data SIAM J. Numer. Anal. 50, pp. 458-467 (10 pages) Online Publication Date: March 15, 2012
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Compressed sensing aims at recovering a sparse signal $x\in \mathbb{R}^N$ from few nonadaptive, linear measurements $\Phi(x)$ given by a measurement matrix $\Phi$. One of the fundamental recovery algorithms is an $\ell_1$ minimization. In this paper we investigate the situation when our measurement $\Phi(x)$ is contaminated by arbitrary noise under the assumption that the matrix $\Phi$ satisfies the restricted isometry property. This complements results from [Candès, Romberg, and Tao, Comm. Pure Appl. Math., 59 (2006), pp. 1207–1223] and [DeVore, Petrova, and Wojtaszczyk, Appl. Comput. Harmon. Anal., 27 (2009), pp. 275–288]. |
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Local A Posteriori Estimates on a Nonconvex Polygonal Domain SIAM J. Numer. Anal. 50, pp. 906-924 (19 pages) Online Publication Date: April 17, 2012
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A class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems on a nonconvex polygonal domain. The results are extensions of previous results for smooth domains [W. Hoffmann et al., Math. Comp., 70 (2001), pp. 897–909; A. H. Schatz and L. B. Wahlbin, Math. Comp., 73 (2004), pp. 517–523]. |
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Composition Methods, Maxwell's Equations, and Source Terms SIAM J. Numer. Anal. 50, pp. 439-457 (19 pages) Online Publication Date: March 13, 2012
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This paper is devoted to high-order numerical time integration of first-order wave equation systems originating from spatial discretization of Maxwell's equations. The focus lies on the accuracy of high-order composition in the presence of source functions. Source functions are known to generate order reduction, and this is most severe for high-order methods. For two methods based on two well-known fourth-order symmetric compositions, convergence results are given assuming simultaneous space-time grid refinement. Herewith physical sources and source functions emanating from Dirichlet boundary conditions are distinguished. Among other things it is shown that the reduction can cost two orders. On the other hand, when a certain perturbation of a source function is used, the reduction is generally diminished by one order. In that case, reduction is absent for physical sources and for Dirichlet sources the order is equal to at least three under stable simultaneous space-time grid refinement. |
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A Nodal-based Finite Element Approximation of the Maxwell Problem Suitable for Singular Solutions SIAM J. Numer. Anal. 50, pp. 398-417 (20 pages) Online Publication Date: March 08, 2012
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A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural $H({\rm \mathbf{curl}}\, 0; \Omega)$ norm for this unknown. In particular, convergence also applies to singular solutions, for which classical nodal-based interpolations are known to suffer from spurious convergence upon mesh refinement. |
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P$_1$-Nonconforming Finite Elements on Triangulations into Triangles and Quadrilaterals SIAM J. Numer. Anal. 50, pp. 418-438 (21 pages) Online Publication Date: March 13, 2012
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The $P_1$-nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected Lipschitz domains. An explicit a priori analysis for the combination of the Park–Sheen and the Crouzeix–Raviart nonconforming finite element methods is given for second-order elliptic PDEs with inhomogeneous Dirichlet boundary conditions. |
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SIAM J. Numer. Anal. 50, pp. 626-642 (17 pages) Online Publication Date: March 29, 2012
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This paper derives some discrete maximum principles for $P1$-conforming finite element approximations for quasi-linear second order elliptic equations. The results are extensions of the classical maximum principles in the theory of partial differential equations to finite element methods. The mathematical tools are based on the variational approach that was commonly used in the classical partial differential equation theory. The discrete maximum principles are established by assuming a property on the discrete variational form that is of global nature. In particular, the assumption on the variational form is verified when the finite element partition satisfies some angle conditions. For the general quasi-linear elliptic equation, these angle conditions indicate that each triangle or tetrahedron needs to be $\mathcal{O}(h^\alpha)$-acute in the sense that each angle $\alpha_{ij}$ (for the triangle) or interior dihedral angle $\alpha_{ij}$ (for the tetrahedron) must satisfy $\alpha_{ij}\le \pi/2-\gamma h^\alpha$ for some $\alpha\ge 0$ and $\gamma>0$. For the Poisson problem where the differential operator is given by the Laplacian, the angle requirement is the same as the existing ones: either all the triangles are nonobtuse or each interior edge is nonnegative. It should be pointed out that the analytical tools used in this paper are based on the powerful De Giorgi iterative method that has played important roles in the theory of partial differential equations. The mathematical analysis itself is of independent interest in the finite element analysis. |
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On the $O(1/n)$ Convergence Rate of the Douglas–Rachford Alternating Direction Method SIAM J. Numer. Anal. 50, pp. 700-709 (10 pages) Online Publication Date: April 10, 2012
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Alternating direction methods (ADMs) have been well studied in the literature, and they have found many efficient applications in various fields. In this note, we focus on the Douglas–Rachford ADM scheme proposed by Glowinski and Marrocco, and we aim at providing a simple approach to estimating its convergence rate in terms of the iteration number. The linearized version of this ADM scheme, which is known as the split inexact Uzawa method in the image processing literature, is also discussed. |
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SIAM J. Numer. Anal. 50, pp. 574-594 (21 pages) Online Publication Date: March 27, 2012
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The coupling of the finite volume element method and the boundary element method is an interesting approach to simulate a coupled system of a diffusion convection reaction process in an interior domain and a diffusion process in the corresponding unbounded exterior domain. This discrete system maintains naturally local conservation, and a possible weighted upwind scheme guarantees the stability of the discrete system also for convection dominated problems. We show existence and uniqueness of the continuous system with appropriate transmission conditions on the coupling boundary, provide a convergence and an a priori analysis in an energy (semi)norm, and provide an existence and an uniqueness result for the discrete system. All results are also valid for the upwind version. Numerical experiments show that our coupling is an efficient method for the numerical treatment of transmission problems, which can also be convection dominated. |
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Linear Rational Finite Differences from Derivatives of Barycentric Rational Interpolants SIAM J. Numer. Anal. 50, pp. 643-656 (14 pages) Online Publication Date: April 04, 2012
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Derivatives of polynomial interpolants lead in a natural way to approximations of derivatives of the interpolated function, e.g., through finite differences. We extend a study of the approximation of derivatives of linear barycentric rational interpolants and present improved finite difference formulas arising from these interpolants. The formulas contain the classical finite differences as a special case and are more stable for calculating one-sided derivatives as well as derivatives close to boundaries. |
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SIAM J. Numer. Anal. 47, pp. 1684-1712 (29 pages) Online Publication Date: April 22, 2009
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We consider scalar conservation laws with the spatially varying flux $H(x)f(u)+(1-H(x))g(u)$, where $H(x)$ is the Heaviside function and $f$ and $g$ are smooth nonlinear functions. Adimurthi, Mishra, and Veerappa Gowda [J. Hyperbolic Differ. Equ., 2 (2005), pp. 783–837] pointed out that such a conservation law admits many $L^1$ contraction semigroups, one for each so-called connection $(A,B)$. Here we define entropy solutions of type $(A,B)$ involving Kružkov-type entropy inequalities that can be adapted to any fixed connection $(A,B)$. It is proved that these entropy inequalities imply the $L^1$ contraction property for $L^\infty$ solutions, in contrast to the “piecewise smooth” setting of Adimurthi, Mishra, and Veerappa Gowda. For a fixed connection, these entropy inequalities include a single adapted entropy of the type used by Audusse and Perthame [Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), pp. 253–265]. We prove convergence of a new difference scheme that approximates entropy solutions of type $(A,B)$ for any connection $(A,B)$ if a few parameters are varied. The scheme relies on a modification of the standard Engquist–Osher flux, is simple as no $2\times2$ Riemann solver is involved, and is designed such that the steady-state solution connecting $A$ to $B$ is preserved. In contrast to most analyses of similar problems, our convergence proof is not based on the singular mapping or compensated compactness methods, but on standard spatial variation estimates away from the flux discontinuity. Some numerical examples are presented. |
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SIAM J. Numer. Anal. 50, pp. 544-573 (30 pages) Online Publication Date: March 27, 2012
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We design arbitrarily high-order accurate entropy stable schemes for systems of conservation laws. The schemes, termed TeCNO schemes, are based on two main ingredients: (i) high-order accurate entropy conservative fluxes and (ii) suitable numerical diffusion operators involving ENO reconstructed cell-interface values of scaled entropy variables. Numerical experiments in one and two space dimensions are presented to illustrate the robust numerical performance of the TeCNO schemes. |
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Convergence of a Particle Method and Global Weak Solutions of a Family of Evolutionary PDEs SIAM J. Numer. Anal. 50, pp. 1-21 (21 pages) Online Publication Date: January 19, 2012
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The purpose of this paper is to provide global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations. The considered family of PDEs is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics. We apply a particle method to the studied evolutionary equations and provide a new self-contained method for proving its convergence. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties. From this result, we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established. |
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SIAM J. Numer. Anal. 50, pp. 710-728 (19 pages) Online Publication Date: April 10, 2012
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In this paper, a rigorous convergence and error analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem is presented. The formulation of the eigenvalue problem in terms of a boundary integral equation yields a nonlinear boundary integral operator eigenvalue problem. This nonlinear eigenvalue problem and its Galerkin approximation are analyzed in the framework of eigenvalue problems for holomorphic Fredholm operator-valued functions. The convergence of the approximation is shown and quasi-optimal error estimates are presented. Numerical experiments are given confirming the theoretical results. |
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A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data SIAM J. Numer. Anal. 45, pp. 1005-1034 (30 pages) Online Publication Date: May 07, 2007
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In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800–825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. |
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SIAM J. Numer. Anal. 50, pp. 669-699 (31 pages) Online Publication Date: April 10, 2012
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This work presents and analyzes a new residual local projection stabilized finite element method (RELP) for the nonlinear incompressible Navier–Stokes equations. Stokes problems defined elementwise drive the construction of the residual-based terms which make the present method stable for the finite element pairs $\mathbb{P}_1/\mathbb{P}_l$, $l=0,1$. Numerical upwinding is incorporated through an extra control on the advective derivative and on the residual of the divergence equation. Well-posedness of the discrete problem as well as optimal error estimates in natural norms are proved under standard assumptions. Next, a divergence-free velocity field is provided by a simple postprocessing of the computed velocity and pressure using the lowest order Raviart–Thomas basis functions. This updated velocity is proved to converge optimally to the exact solution. Numerics assess the theoretical results and validate the RELP method. |
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Finite Element Method for a Nonlocal Problem of Kirchhoff Type SIAM J. Numer. Anal. 50, pp. 657-668 (12 pages) Online Publication Date: April 10, 2012
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The nonlocal term in the nonlinear equations of Kirchhoff type causes difficulties when the equation is solved numerically by using the Newton–Raphson method. This is because the Jacobian of the Newton–Raphson method is full. In this article, the finite element system is replaced by an equivalent system for which the Jacobian is sparse. We derive quasi-optimal error estimates for the finite element method and demonstrate the results with numerical experiments. |
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Monotone Piecewise Cubic Interpolation SIAM J. Numer. Anal. 17, pp. 238-246 (9 pages) Online Publication Date: July 17, 2006
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Necessary and sufficient conditions are derived for a cubic to be monotone on an interval. These conditions are used to develop an algorithm which constructs a visually pleasing monotone piecewise cubic interpolant to monotone data. Several examples are given which compare this algorithm with other interpolation methods. |
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A Local Entropy Minimum Principle for Deriving Entropy Preserving Schemes SIAM J. Numer. Anal. 50, pp. 468-491 (24 pages) Online Publication Date: March 20, 2012
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The present work deals with the establishment of stability conditions of finite volume methods to approximate weak solutions of the general Euler equations to simulate compressible flows. In order to ensure discrete entropy inequalities, we derive a new technique based on a local minimum principle to be satisfied by the specific entropy. Sufficient conditions are exhibited to satisfy the required local minimum entropy principle. Arguing these conditions, a class of entropy preserving schemes is thus derived. |
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Mixed Finite Element Method for a Degenerate Convex Variational Problem from Topology Optimization SIAM J. Numer. Anal. 50, pp. 522-543 (22 pages) Online Publication Date: March 22, 2012
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The optimal design task of this paper seeks the distribution of two materials of prescribed amounts for maximal torsion stiffness of an infinite bar of a given cross section. This example of relaxation in topology optimization leads to a degenerate convex minimization problem $E\left( v \right):= \int_\Omega \varphi_0\left( \left\lvert\nabla v\right\rvert \right)\operatorname{dx} - \int_\Omega fv\operatorname{dx}\text{for } v \in V:=H^1_0( \Omega )$ with possibly multiple primal solutions $u$, but with unique stress $\sigma:=\varphi_0'\left( \left\lvert\nabla u\right\rvert \right)\operatorname{sign}\nabla u.$ The mixed finite element method is motivated by the smoothness of the stress variable $\sigma \in H^1_{\operatorname{loc}}( \Omega ; \mathbb R^2)$, while the primal variables are uncontrollable and possibly nonunique. The corresponding nonlinear mixed finite element method is introduced, analyzed, and implemented. The striking result of this paper is a sharp a posteriori error estimation in the dual formulation, while the a posteriori error analysis in the primal problem suffers from the reliability-efficiency gap. An empirical comparison of that primal formulation with the new mixed discretization schemes is intended for uniform and adaptive mesh refinements. |
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