Error estimates for the numerical approximation of Neumann control problems

We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193–219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order O(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L ²(Γ). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45–61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided. The authors were supported by Ministerio de Ciencia y Tecnología (Spain).     

Bifurcation problems for a class of degenerate quasilinear elliptic equations

In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.     

Gradient estimates for a class of quasilinear elliptic equations with measure data

We study a class of non-homogeneous quasilinear elliptic equations with measure data to obtain an optimal regularity estimate. We prove that the gradient of a weak solution to the problem is as integrable as the first order maximal function of the associated measure in the Orlicz spaces up to a correct power.     

On the structure of solutions to a class of quasilinear elliptic Neumann problems

We study the structure of positive solutions to the equation ?^mΔ_mu-u^m-1+f(u)=0 with homogeneous Neumann boundary condition. First, we show the existence of a mountain-pass solution and find that as ?→0⁺ the mountain-pass solution develops into a spike-layer solution. Second, we prove that there is an uniform upper bound independent of ? for any positive solution to our problem. We also present a Harnack-type inequality for the positive solutions. Finally, we show that if 1<m?2 holds and ? is sufficiently large, any positive solution must be a constant.     

Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations

In this paper, we investigate the L ^??(L ₂)-error estimates and superconvergence of the semidiscrete mixed finite elementmethods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k ?? 0). We derive error estimates for approximation of both state and control. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problems.     

Error estimates for the approximation of multibang control problems

In the present work we apply an augmented Lagrange method to solve pointwise state constrained elliptic optimal control problems. We prove strong convergence of the primal variables as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. In addition, we show that the sequence of generated penalty parameters is bounded only in exceptional situations, which is different from classical results in finite-dimensional optimization. In addition, numerical results are presented.     

A posteriori error estimates for optimal control problems governed by parabolic equations

Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980     

加权奇异拟线性椭圆方程特征值问题

在加权Sobolev空间中,利用Galerkin方法及推广的Brouwer定理,研究一类奇异拟线性椭圆方程高阶特征问题,得到了其非平凡弱解的存在性.     

Error estimates of expanded mixed methods for optimal control problems governed by hyperbolic integro‐differential equations

In this article, we investigate the L^∞(L²) ‐error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart‐Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k(k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Finite-difference discretizations of quadratic control problems governed by ordinary elliptic differential equations

In this paper, we analyze finite difference discretizations for a class of control constrained elliptic optimal control problems. If the optimal control has a derivative of bounded variation, we show discrete quadratic convergence in terms of the mesh size h of the discrete optimal controls. Furthermore, based on the optimality conditions, we construct a new discrete control for which we derive continuous error estimates of order h ².     

Error estimates for the numerical approximation of a quaslinear Neumann problem under minimal regularity of the data

The finite element based approximation of a quasilinear elliptic equation of non monotone type with Neumann boundary conditions is studied. Minimal regularity assumptions on the data are imposed. The consideration is restricted to polygonal domains of dimension two and polyhedral domains of dimension three. Finite elements of degree k ≥ 1 are used to approximate the equation. Error estimates are established in the L ²(Ω) and H ¹(Ω) norms for convex and non-convex domains. The issue of uniqueness of a solution to the approximate discrete equation is also addressed.     

Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations

In this paper, we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by Stokes equations. We obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods, which are the patch recovery technique and the least-squares surface fitting method. Our results are based on some regularity assumption for the Stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions.     

On the structure of solutions to a class of quasilinear elliptic Neumann problems. Part II

We continue our work (Y. Li, C. Zhao in J Differ Equ 212:208–233, 2005) to study the structure of positive solutions to the equation ε ^m Δ _m u ? u ^m-1 + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of ${\mathbb{R}^{N}\,\left(N\geq 2\right)}$ . First, we study subcritical case for 2 < m < N and show that after passing by a sequence positive solutions go to a constant in C ^1,α sense as ε → ∞. Second, we study the critical case for 1 < m < N and prove that there is a uniform upper bound independent of ${\varepsilon \in \lbrack 1,\infty)}$ for the least-energy solutions. Third, we show that in the critical case for 1 < m ≤ 2 the least energy solutions must be a constant if ε is sufficiently large and for 2 < m < N the least energy solutions go to a constant in C ^1,α sense as ε → ∞.     

Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form

Summary Conditions are given for the validity of long range quasi optimal error estimates of the Galerkin method for quasilinear parabolic equations. If the corresponding nonlinear elliptic operator satisfies a coercivity condition, the stationary problem may be approximated by the numerical solution of the parabolic problem when the time variablet is large enough, starting with an arbitrary initial function. Some practical applications are discussed.