L ∞ stability of the MUSCL methods

We present a general L ^∞ stability result for generic finite volume methods coupled with a large class of reconstruction for hyperbolic scalar equations. We show that the stability is obtained if the reconstruction respects two fundamental properties: the convexity property and the sign inversion property. We also introduce a new MUSCL technique named the multislope MUSCL technique based on the approximations of the directional derivatives in contrast to the classical piecewise reconstruction, the so-called monoslope MUSCL technique, based on the gradient reconstruction. We show that under specific constraints we shall detail, the two MUSCL reconstructions satisfy the convexity and sign inversion properties and we prove the L ^∞ stability.     

Analytic Solutions of L∞ Optimal Control Problems for the Wave Equation

There are very few results about analytic solutions of problems of optimal control with minimal L norm. In this paper, we consider such a problem for the wave equation, where the derivative of the state is controlled at both boundaries. We start in the zero position and consider a problem of exact control, that is, we want to reach a given terminal state in a given finite time. Our aim is to find a control with minimal L norm that steers the system to the target.We give the analytic solution for certain classes of target points, for example, target points that are given by constant functions. For such targets with zero velocity, the analytic solution has been given by Bennighof and Boucher in Ref. 1.     

Optimal L∞-error estimates for nonconforming and mixed finite element methods of lowest order

Summary For second order linear elliptic problems, it is proved that theP ₁-nonconforming finite element method has the sameL -asymptotic accuracy as theP ₁-conforming one. This result is applied to derive optimalL -error estimates for both the displacement and the stress fields of the lowest order Raviart-Thomas mixed finite element method, and a superconvergence result at the barycenter of each element.Performed in the research program of Istituto di Analisi Numerica of C.N.R. of PaviaPartially supported by MPI, GNIM of CNR, ItalySupported by Consejo Nacional de Investigaciones Cientificas y Técnicas, Argentina     

Optimal l~∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions

Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h~2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.     

L ∞ andW 1, ∞ estimates for second order generalized difference methods on two-point boundary value problems

In this paper, we consider the second order generalized difference scheme for the two-point boundary value problem and obtain optimal order error estimates inL ^∞ andW ^{1, ∞}. The results in this paper perfect the theory of the second order generalized difference method.     

L∞-Estimates for the Torsion Function and L∞-Growth of Semigroups Satisfying Gaussian Bounds

Potential Analysis - We investigate selfadjoint C0-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic...     

An Optimal L∞–error Estimate for an Approximation of a Parabolic Variational Inequality

In this article, an optimal error estimate for parabolic variational inequalities is studied. Existence and uniqueness of the solution is provided by the introduction of a constructive algorithm. An optimally L^∞-asymptotic behavior in uniform norm is proved using the semi-implicit time scheme combined with the finite element spatial approximation. The approach is based on the concept of subsolutions.     

A Note on the Choice of Optimal Scores for Ordinal Data

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Estimates for the difference between exact and approximate solutions of parabolic equations on the basis of Poincaré inequalities for traces of functions on the boundary

We study a method for the derivation of majorants for the distance between the exact solution of an initial–boundary value reaction–convection–diffusion problem of the parabolic type and an arbitrary function in the corresponding energy class. We obtain an estimate (for the deviation from the exact solution) of a new type with the use of a maximally broad set of admissible fluxes. In the definition of this set, the requirement of pointwise continuity of normal components of the dual variable (which was a necessary condition in earlier-obtained estimates) is replaced by the requirement of continuity in the weak (integral) sense. This result can be achieved with the use of the domain decomposition and special embedding inequalities for functions with zero mean on part of the boundary or for functions with the zero mean over the entire domain.     

B-Theory of Runge-Kutta methods for stiff Volterra functional differential equations

B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations(VFDEs)are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems(IVPs)in ordinary differentialequations(ODEs),delay differential equations(DDEs),integro-differential equatioons(IDEs)and VFDEs of     

Poisson kernel is the unique approximate identity,asymptotically multiplicative on H∞

Let for anyf H(R), where (x): = ^–1(x^–1). Then (x) P (x + h) for some h R and > 0; P denotes the Poisson kernel.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 82–89, 1989.     

Optimal error estimate of the local discontinuous Galerkin methods based on the generalized alternating numerical fluxes for nonlinear convection–diffusion equations

Numerical Algorithms - In this paper, we present the optimal L2-norm error estimate of the local discontinuous Galerkin method based on the generalized alternating numerical flux for nonlinear...     

On the Lévy Measures for Symmetric Stable Semigroups on the Torus T∞

We investigate analytical properties of the Lévy measures fir symmetric stable semigroups on the compact Lie-projective group T. We apply these properties to describe the domain of the fractional powers of Laplacians on T. Among the analytical tools involved are the intrinsic metric and the scale of Hölder continuous functions w.r.t. this metric.     

L∞-boundedness of the finite element galerkin operator for parabolic problems

In this paper the heat equation with Dirichlet boundary conditions in N ≤ 3 space dimensions - serving as model problem of second order parabolic initial boundary value problems - is considered. We prove: The standard finite element method is uniformly bounded in L_∞ with respect to space and time if the underlying finite elements are at least cubics.     

Optimal L ∞-Error Estimate of a Finite Element Method for Hamilton–Jacobi–Bellman Equations

This paper is concerned with the piecewise linear finite element approximation of Hamilton–Jacobi–Bellman equations. We establish the optimal L ^∞-error estimate, combining the concepts of subsolution and discrete regularity.     

L∞-error estimate for the numerical treatment of the obstacle problem by the penalty method

The penalty method is used to compute approximations of the solution of the obstacle problem and error estimates in the L_∞-norm are derived. The error is of the same order asforthe corresponding unrestricted problem.