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## Priori Estimates (priori + estimate)
## Selected Abstracts## A robust a priori error estimate for the Fortin,Soulie finite element method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2005David J. BlackerArticle first published online: 14 MAR 200Abstract It is well known that conforming finite element schemes exhibit Poisson locking in the incompressible limit as the Poisson ratio , tends to 1/2. A remedy for this is to use a non-conforming method (Math. Comput. 1992; 59:321-328) in which an a priori error bound is proved for the Crouzeix,Raviart scheme. In this paper we derive a new a priori estimate for the error in energy for the Fortin,Soulie finite element method using a method similar to that used in Brenner and Sung. We then illustrate the new error bound by presenting some numerical examples. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluid MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2003J. L. FerrinWe undertake a rigorous derivation of the Biot's law for a porous elastic solid containing an inviscid fluid. We consider small displacements of a linear elastic solid being itself a connected periodic skeleton containing a pore structure of the characteristic size ,. It is completely saturated by an incompressible inviscid fluid. The model is described by the equations of the linear elasticity coupled with the linearized incompressible Euler system. We study the homogenization limit when the pore size ,tends to zero. The main difficulty is obtaining an a priori estimate for the gradient of the fluid velocity in the pore structure. Under the assumption that the solid part is connected and using results on the first order elliptic systems, we obtain the required estimate. It allows us to apply appropriate results from the 2-scale convergence. Then it is proved that the microscopic displacements and the fluid pressure converge in 2-scales towards a linear hyperbolic system for an effective displacement and an effective pressure field. Using correctors, we also give a strong convergence result. The obtained system is then compared with the Biot's law. It is found that there is a constitutive relation linking the effective pressure with the divergences of the effective fluid and solid displacements. Then we prove that the homogenized model coincides with the Biot's equations but with the added mass ,a being a matrix, which is calculated through an auxiliary problem in the periodic cell for the tortuosity. Furthermore, we get formulas for the matricial coefficients in the Biot's effective stress,strain relations. Finally, we consider the degenerate case when the fluid part is not connected and obtain Biot's model with the relative fluid displacement equal to zero. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Oscillatory asymptotic expansion for semilinear wave equation MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2001Stefania Di PomponioWe study oscillatory properties of the solution to semilinear wave equation, assuming oscillatory terms in initial data have sufficiently small amplitude. The main result gives an a priori estimate of the remainder in the approximation by means of the method of geometric optics. The method of establishing this estimate is based on a combination between energy type estimates for transport equation and Sobolev embedding. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## A remark on the coercivity for a first-order least-squares method NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2007Jaeun KuAbstract This study present a short proof concerning the coercivity of a first-order least-squares finite element method for general second-order elliptic problems proposed by Cai, Lazarov, Manteuffel and McCormick (Cai et al. J Numer Anal 31 (1994), 1785,1799). Our proof is based on a priori estimate and the technique can be applied to prove L2 -norm error estimate for the primary function u. After establishing the coercivity bound from the assumed a priori estimate, we observe that the coercivity bound is actually equivalent to the a priori estimate. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] ## Error analysis of the L2 least-squares finite element method for incompressible inviscid rotational flows, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2004Chiung-Chiou TsaiAbstract In this article we analyze the L2 least-squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity-vorticity-pressure formulation. The least-squares functional is defined in terms of the sum of the squared L2 norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first-order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H1 norm for velocity and pressure and a suboptimal rate of convergence in the L2 norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 [source] ## Centrosymmetric and pseudo-centrosymmetric structures refined as non-centrosymmetric ACTA CRYSTALLOGRAPHICA SECTION B, Issue 5 2006H. D. FlackThe behaviour of the Flack parameter for centrosymmetric and pseudo-centrosymmetric crystal structures based on crystal structures published as being non-centrosymmetric is presented. It is confirmed for centrosymmetric structures that the value obtained for the Flack parameter is critically dependent on the Friedel coverage of the intensity data, approaching 0.5 for a coverage of 100% and sticking near the starting value for a coverage of 0%. For pseudo-centrosymmetric structures, even those very close to being centrosymmetric, it is found that it is often possible to obtain significant values of the Flack parameter. A theoretical basis for this surprising result is established. It has also been possible to establish an a priori estimate of the standard uncertainty of the Flack parameter based only on the chemical composition of the compound and the wavelength of the radiation. The paper concludes with preliminary presentations of bias in the Flack parameter and of inconsistent chemical and crystallographic data. [source] ## A priori estimate for convex solutions to special Lagrangian equations and its application COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 4 2009Jingyi ChenWe derive a priori interior Hessian estimates for special Lagrangian equations when the potential is convex. When the phase is very large, we show that continuous viscosity solutions are smooth in the interior of the domain. © 2008 Wiley Periodicals, Inc. [source] ## First-Order Schemes in the Numerical Quantization Method MATHEMATICAL FINANCE, Issue 1 2003V. BallyThe numerical quantization method is a grid method that relies on the approximation of the solution to a nonlinear problem by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first-order scheme based on piecewise linear approximations. Main ingredients are correction terms in the transition probability weights. We emphasize the fact that in the case of optimal quantization, many of these correcting terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and a priori estimates of the errors are proposed. [source] ## Existence of solutions to a phase transition model with microscopic movements MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2009Eduard FeireislAbstract We prove the existence of weak solutions for a 3D phase change model introduced by Michel Frémond in (Non-smooth Thermomechanics. Springer: Berlin, 2002) showing, via a priori estimates, the weak sequential stability property in the sense already used by the first author in (Comput. Math. Appl. 2007; 53:461,490). The result follows by passing to the limit in an approximate problem obtained adding a superlinear part (in terms of the gradient of the temperature) in the heat flux law. We first prove well posedness for this last problem and then,using proper a priori estimates,we pass to the limit showing that the total energy is conserved during the evolution process and proving the non-negativity of the entropy production rate in a suitable sense. Finally, these weak solutions turn out to be the classical solution to the original Frémond's model provided all quantities in question are smooth enough. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## A weighted Lq -approach to Oseen flow around a rotating body MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2008R. FarwigAbstract We study time-periodic Oseen flows past a rotating body in ,3 proving weighted a priori estimates in Lq -spaces using Muckenhoupt weights. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional terms (,,,,x),,,,,u and ,,,,,u in the equation of momentum where , denotes the angular velocity. Due to the asymmetry of Oseen flow and to describe its wake we use anisotropic Muckenhoupt weights, a weighted theory of Littlewood,Paley decomposition and of maximal operators as well as one-sided univariate weights, one-sided maximal operators and a new version of Jones' factorization theorem. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## On the structural stability of thermoelastic model of porous media MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2008Stan ChiriAbstract In the present paper we study the structural stability of the mathematical model of the linear thermoelastic materials with voids. We prove that the solutions of problems depend continuously on the constitutive quantities, which may be subjected to error or perturbations in the mathematical modelling process. Thus, we assume to have changes in the various coupling coefficients of the model and then we establish estimates of continuous dependence of solutions. We have to outline that such estimates play a central role in obtaining approximations to these kinds of problems. To derive a priori estimates for a solution we first establish appropriate bounds for the solutions of certain auxiliary problems. These are achieved by means of so-called Rellich-like identities. We also investigate how the solution in the coupled model behaves as some coupling coefficients tend to zero. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Boundary value problem for the N -dimensional time periodic Vlasov,Poisson system MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2006M. BostanAbstract In this work, we study the existence of time periodic weak solution for the N -dimensional Vlasov,Poisson system with boundary conditions. We start by constructing time periodic solutions with compact support in momentum and bounded electric field for a regularized system. Then, the a priori estimates follow by computations involving the conservation laws of mass, momentum and energy. One of the key point is to impose a geometric hypothesis on the domain: we suppose that its boundary is strictly star-shaped with respect to some point of the domain. These results apply for both classical or relativistic case and for systems with several species of particles. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Continuous dependence on modelling for a complex Ginzburg,Landau equation with complex coefficients MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2004Yongfu YangAbstract Continuous dependence on a modelling parameter is established for solutions of a problem for a complex Ginzburg,Landau equation. A homogenizing boundary condition is also used to discuss the continuous dependence results. We derive a priori estimates that indicate that solutions depend continuously on a parameter in the governing differential equation. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Non-linear initial boundary value problems of hyperbolic,parabolic type. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002A general investigation of admissible couplings between systems of higher order.This is the second part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so-called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right-hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non-linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Non-linear initial boundary value problems of hyperbolic,parabolic type. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002A general investigation of admissible couplings between systems of higher order.This is the third part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so-called energy method. In the above sense the regularity assumptions about the coefficients and right-hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Global continuation for first order systems over the half-line involving parameters MATHEMATISCHE NACHRICHTEN, Issue 8 2009Gilles EvéquozArticle first published online: 21 JUL 200Abstract Let X be one of the functional spaces W1,p ((0, ,), ,N) or C01 ([0, ,), ,N), we study the global continuation in , for solutions (,, u, ,) , , × X × ,k of the following system of ordinary differential equations: where ,N = X1 , X2 is a given decomposition, with associated projection P: ,N , X1. Under appropriate conditions upon the given functions F and ,, this problem gives rise to a nonlinear Fredholm operator which is proper on the closed bounded subsets of , × X × ,k and whose zeros correspond to the solutions of the original problem. Using a new abstract continuation result, based on a recent degree theory for proper Fredholm mappings of index zero, we reduce the continuation problem to that of finding a priori estimates for the possible solutions (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Finite element approximation of a forward and backward anisotropic diffusion model in image denoising and form generalization NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2008Carsten EbmeyerAbstract A new forward,backward anisotropic diffusion model is introduced. The two limit cases are the Perona-Malik equation and the Total Variation flow model. A fully discrete finite element scheme is studied using C0 -piecewise linear elements in space and the backward Euler difference scheme in time. A priori estimates are proven. Numerical results in image denoising and form generalization are presented.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] ## A multilevel finite element method in space-time for the Navier-Stokes problem, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2005Yinnian HeAbstract A multilevel finite element method in space-time for the two-dimensional nonstationary Navier-Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier-Stokes problem is only solved on a single coarsest space-time mesh; subsequent approximations are generated on a succession of refined space-time meshes by solving a linearized Navier-Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J -level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: hj , h, kj , k, j = 2, ,, J, the J -level finite element method in space-time provides the same accuracy as the one-level method in space-time in which the fully nonlinear Navier-Stokes problem is solved on a final finest space-time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] ## A priori estimates for fluid interface problems COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2008Jalal ShatahWe consider the regularity of an interface between two incompressible and inviscid fluid flows in the presence of surface tension. We obtain local-in-time estimates on the interface in H(3/2)k + 1 and the velocity fields in H(3/2)k. These estimates are obtained using geometric considerations which show that the Kelvin-Helmholtz instabilities are a consequence of a curvature calculation. © 2007 Wiley Periodicals, Inc. [source] ## Geometry and a priori estimates for free boundary problems of the Euler's equation COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2008Jalal ShatahIn this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to 0, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension. © 2007 Wiley Periodicals, Inc. [source] |