Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established and some numerical experiments, which confirm the theoretical results, are performed.The first two authors were supported by Ministerio de Ciencia y Tecnología (Spain). The second author was also supported by the DFG research center “Mathematics for key technologies” (FZT86) in Berlin.     

Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

In this paper, we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions. We derive some superconvergence properties for the control variable and the state variables. Moreover, we derive $L^∞$- and $H^{-1}$-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.     

A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems

In this paper, a constrained distributed optimal control problem governed by a first-order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient (PCG) and algebraic multi-grid (AMG) can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in $L^2(Ω)$-norm, for the original state and adjoint state in $H^1(Ω)$-norm, and for the flux state and adjoint flux state in $H$(div; $Ω$)-norm. Finally, we use one numerical example to validate the theoretical findings.     

Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.     

On Mixed Error Estimates for Elliptic Obstacle Problems

We establish in this paper sharp error estimates of residual type for finite element approximation to elliptic obstacle problems. The estimates are of mixed nature, which are neither of a pure a priori form nor of a pure a posteriori form but instead they are combined by an a priori part and an a posteriori part. The key ingredient in our derivation for the mixed error estimates is the use of a new interpolator which enables us to eliminate inactive data from the error estimators. One application of our mixed error estimates is to construct a posteriori error indicators reliable and efficient up to higher order terms, and these indicators are useful in mesh-refinements and adaptive grid generations. In particular, by approximating the a priori part with some a posteriori quantities we can successfully track the free boundary for elliptic obstacle problems.     

半线性椭圆型问题Mortar有限元逼近的瀑布型多重网格法

Mortar有限元法作为一个非协调的区域分解技术已得到许多研究者的关注(如文献[2]、[5]等)。本文对半线性椭圆型问题的Mortar有限元逼近提出了瀑布型多重网格法,并给出了此法的误差估计和计算复杂度估计定理。     

Elliptic Reconstruction and a Posteriori Error Estimates for Fully Discrete Semilinear Parabolic Optimal Control Problems

This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the $L^\infty(0,T;L^2(\Omega))$-norm. Finally, a numerical experiment is performed to illustrate the performance of the derived estimators.     

A Posteriori Error Estimates for Distributed Convex Optimal Control Problems

In this paper, we present an a posteriori error analysis for finite element approximation of distributed convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximation schemes for control problems. Explicit estimates are obtained for some model problems which frequently appear in real-life applications.     

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.     

On the Error Estimates for the Finite Element Approximation of a Class of Boundary Optimal Control Systems

In this article, we consider an application of the abstract error estimate for a class of optimal control systems described by a linear partial differential equation (as stated in Numer. Funct. Anal. Optim. 2009; 30:523–547). The control is applied at the boundary and we consider both, Neumann and Dirichlet optimal control problems. Finite element methods are proposed to approximate the optimal control considering an approximation of the variational inequality resulting from the optimality conditions; this approach is known as classical one. We obtain optimal order error estimates for the control variable and numerical examples, taken from the literature, are included to illustrate the results.     

On the Uniqueness of the Positive Solution for a Semilinear Elliptic Problem with Critical Exponent

Inrecentyears,thequestionsofexistence,uniquenessandmultiplicityofsolutionstothe.problemhasbeenmuchinvestigated(see[l-ll]).WhereOisasmoothboundedregioninRN,N23and6istheLaplacianoperator.Thesolvabilityofboundaryvalueproblemdependsonthegrowthofthenonlinearityf(u)andontheregion,boundedorunbounded,unitballoranyotherregions.Wesaythatfunctionf(u)growssubcriticallyifthereexistsqe(l,'(N 2)/(N-2))suchthatliT32PIf(u)l/lulq相似文献   

Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L₂ and H 1 . The convergence rate in the L_∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

一种非线性对流扩散方程的特征有限元方法的误差估计

给出求解一种二维非线性对流扩散方程组的 Grank-Nicolson型特征有限元方法 ,并给出该方法的 H1模最优阶误差估计 .     

Gaussian Curvature Estimates for the Convex Level Sets of Solutions for Some Nonlinear Elliptic Partial Differential Equations

We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary.     

带梯度项的椭圆型方程边界爆破解的高阶估计

In this paper,the higher order asymptotic behaviors of boundary blow-up solutions to the equation■in bounded smooth domain■are systematically investigated for p and q.The second and third order boundary behaviours of the equation are derived.The results show the role of the mean curvature of the boundary■and its gradient in the high order asymptotic expansions of the solutions.     

Existence of Entire Solutions of a Singular Semilinear Elliptic Problem

In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang‘s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.