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.     

An Estimate of Inequality of Signature and Nullity for Links

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Optimal Control Problems for Elliptic–Parabolic Variational Inequalities with Time-Dependent Constraints

In this paper, we study optimal control problems for quasi-linear elliptic–parabolic variational inequalities with time-dependent constraints. We prove the existence of an optimal control that minimizes the nonlinear cost functional. Moreover, we apply our general results to some model problems. In particular, we show the necessary condition of optimal pair for a problem of partial differential equation (PDE) with a non-homogeneous Dirichlet boundary condition.     

An Optimal Control Problem of a Coupled Nonlinear Parabolic Population System

An optimal control problem for a coupled nonlinear parabolic population system is considered. The existence and uniqueness of the positive solution for the system is shown by the method of upper and lower solutions. An explicit prior bound of solutions to the system is given by considering an auxiliary coupled linear system. The existence of the optimal control is proved and the characterization of the optimal control is established.     

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

We derive a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge–Kutta Collocation (RK-C) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ODEs and parabolic PDEs. The key ingredients in deriving these bounds are appropriate one-degree higher continuous reconstructions of the approximate solutions and pointwise error representations. The reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time-step; they do not hinge on asymptotics.     

Optimal Control of Elliptic Variational–Hemivariational Inequalities

This paper establishes a bridge between set optimization problems and vector Ky Fan inequality problems. We introduce a general model, called the bifunction-set optimization problem, that provides a unifying framework for the above-mentioned problems. An existence result in our model is obtained, with the help of KKM–Fan’s lemma. As applications, we derive some new or sharper existence results for set optimization problems and generalized vector Ky Fan inequalities with efficient solutions.     

An a Posteriori Error Estimate for a Semi-Lagrangian Scheme for Hamilton–Jacobi Equations

We present an a posteriori estimate for a first order semi-Lagrangian method for Hamilton–Jacobi equations. The result requires piecewise C ^1,1 regularity of the viscosity solution and is stated for the Bellman equation related to the infinite horizon problem, although it can be applied to more general Hamilton–Jacobi equations with convex Hamiltonians. This estimate suggests different numerical indicators that can be used to construct an adaptive algorithm for the approximation of the viscosity solution.     

L ∞-Error estimate for an approximation of a parabolic variational inequality

Summary Almost optimalL -convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase Stefan problem. The discretization employs piecewise linear finite elements in space and the backward Euler scheme in time. By means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation. The latter bound is obtained by using the smoothing property of the Galerkin method.     

An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality

We prove a sharp Alexandrov–Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic n-space, n ≥ 3. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter–Schwarzschild solution. This sharpens previous results by Dahl–Gicquaud–Sakovich and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space–times with negative cosmological constant. We also explain how our methods can be easily adapted to derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension n ≥ 3. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chru?ciel and Simon on the validity of a Penrose-type inequality for exotic black holes.     

Variational Approximation for Fokker–Planck Equation on Riemannian Manifold

Under the bounded geometry assumption on Riemannian manifold M, a variational approximation for Fokker–Planck equation on M is constructed by the scheme of Jordan et al. in SIAM J Math Anal 29(1):1–17, 1998. Moreover, the uniqueness and global L ^p -estimate of the solution for 1 < p < dim(M)/(dim(M) ? 1) are obtained for a broad class of potential.     

An Optimal Control Problem for a Singular System of Solid–Liquid Phase Transition

In this article, we deal with a control problem for a singular system regarding a phase-field model which describes a solid–liquid transition by the Ginzburg–Landau theory. The purpose is to control the system by the means of the heat supply r able to guide it into a certain state with a solid (or liquid) part in a prescribed subset Ω₀ of the space domain Ω, and maintain it in this state during a period of time. The transition is described by a nonlinear differential system of two equations for the phase field and temperature. The control problem is set for some expressions of the cost functional which might reveal cases of physical interest. An approximating control problem is introduced and the existence of at least an optimal pair is proved. The first-order optimality conditions for the approximating problem are determined and a convergence result is given.     

Variational inequalities for a system of elliptic–parabolic equations

We create a general framework for mathematical study of variational inequalities for a system of elliptic–parabolic equations. In this paper, we establish a solvability theorem concerning the existence of solutions for the vector-valued elliptic–parabolic variational inequality with time-dependent constraint. Moreover, we give some applications of the system, for example, time-dependent boundary obstacle problem and time-dependent interior obstacle problem.     

Optimal point-wise error estimate of a compact difference scheme for the Klein–Gordon–Schrödinger equation

In this paper, we propose a compact finite difference scheme for computing the Klein–Gordon–Schrödinger equation (KGSE) with homogeneous Dirichlet boundary conditions. The proposed scheme not only conserves the total mass and energy in the discrete level but also is linearized in practical computation. Except for the standard energy method, a new technique is introduced to obtain the optimal convergent rate, without any restriction on the grid ratios, at the order of O(h⁴+τ²)$O(h^4+{\tau }^2)$ in the l^∞$l^{\infty }$-norm with time step τ and mesh size h. Finally, numerical results are reported to test the theoretical results.     

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.     

On Weak Solutions of a Class of Variational Inequalities of Parabolic Type7

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Optimal error estimates for the pseudostress formulation of the Navier–Stokes equations

In this paper, we prove optimal a priori error estimates for the pseudostress-velocity mixed finite element formulation of the incompressible Navier–Stokes equations, thus improve the result of Cai et al. (SINUM 2010). This is achieved by applying Petrov–Galerkin type Brezzi–Rappaz–Raviart theory.     

Verified computation of solutions for obstacle problems with guaranteed L∞ error bound

In this paper, we consider a numerical enclosure method with guaranteed L^∞ error bound for the solutions of obstacle problems. Using the finite-element approximations and the explicit a priori error estimates for obstacle problems, we present an effective verification procedure that automatically generates on a computer a set which includes the exact solution. A particular emphasis is that our method needs no assumption of the existence of the solution of the original obstacle problems, but it follows as the result of computation itself. A numerical example for an obstacle problem is presented.