New a posteriori $L^{\infty }(L^2) $ and $L^2(L^2)$-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

Zuliang Lu

Displaying similar documents to “New a posteriori $L^{\infty} (L^{2})$ and $L^{2} (L^{2})$ -error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems”

Finite element analysis for a regularized variational inequality of the second kind

Zhang, Tie, Zhang, Shuhua, Azari, Hossein

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In this paper, we investigate the a priori and the a posteriori error analysis for the finite element approximation to a regularization version of the variational inequality of the second kind. We prove the abstract optimal error estimates in the $H^{1}$ - and $L_{2}$ -norms, respectively, and also derive the optimal order error estimate in the $L_{\infty}$ -norm under the strongly regular triangulation condition. Moreover, some residual–based a posteriori error estimators are established, which can provide the...

An observability estimate for parabolic equations from a measurable set in time and its applications

Kim Dang Phung, Gengsheng Wang (2013)

Journal of the European Mathematical Society

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This paper presents a new observability estimate for parabolic equations in $Ω \times (0, T)$ , where $Ω$ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $Ω$ and a subset of positive measure in $(0, T)$ . This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

A note on control of the false discovery proportion

Marcin Dudziński, Konrad Furmańczyk (2009)

Applicationes Mathematicae

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We consider the problem of simultaneous testing of a finite number of null hypotheses $H_{i}$ , i=1,...,s. Starting from the classical paper of Lehmann (1957), it has become a very popular subject of research. In many applications, particularly in molecular biology (see e.g. Dudoit et al. (2003), Pollard et al. (2005)), the number s, i.e. the number of tested hypotheses, is large and the popular procedures that control the familywise error rate (FWERM) have small power. Therefore, we are concerned...

Synthesis of optimal control for nonlinear third order systems

Wiesław Szwiec

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The paper is devoted to a certain problem in the theory of synthesis of optimal control. The fundamental results of research in the synthesis of optimal control for second order systems are given in [1], [2], [5], [10]. In particular, in [2] and [5] some specific nonlinear systems are investigated, namely the systems related with the differential equations of the forms̈ + F(s,ṡ) = u or s̈ + F(s,ṡ,u) = 0where u is a control parameter in the interval [—1,1]. Some results concerning third...

Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n)

Olivier Ramaré (2013)

Acta Arithmetica

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We prove that the error term $\sum_{n \leq x} Λ (n) / n - l o g x + γ$ differs from (ψ(x)-x)/x by a well controlled function. We deduce very precise numerical results from the formula obtained.

Stability and sensitivity analysis for optimal control problems with control-state constraints

Kazimierz Malanowski

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A family of parameter dependent optimal control problems ${(O)}_{h}$ with smooth data for nonlinear ODEs is considered. The problems are subject to pointwise mixed control-state constraints. It is assumed that, for a reference value h₀ of the parameter, a solution of ${(O)}_{h ₀}$ exists. It is shown that if (i) independence, controllability and coercivity conditions are satisfied at the reference solution, then (ii) for each h from a neighborhood of h₀, a locally unique solution to ${(O)}_{h}$ and the associated Lagrange...

Space-time adaptive $h p$ -FEM: Methodology overview

Šolín, Pavel, Segeth, Karel, Doležel, Ivo

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We present a new class of self-adaptive higher-order finite element methods ( $h p$ -FEM) which are free of analytical error estimates and thus work equally well for virtually all PDE problems ranging from simple linear elliptic equations to complex time-dependent nonlinear multiphysics coupled problems. The methods do not contain any tuning parameters and work reliably with both low- and high-order finite elements. The methodology was used to solve various types of problems including thermoelasticity,...

The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Tie Zhu Zhang, Shu Hua Zhang (2015)

Applications of Mathematics

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We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$ -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$ , the gradient approximation possesses the superconvergence: ${max}_{P \in S} | (\nabla u - \bar{\nabla} u_{h}) (P) | = O (h^{2}) {| ln h |}^{3 / 2}$ , where $\bar{\nabla}$ denotes the average gradient on elements containing vertex $P$ . Furthermore, by using the interpolation post-processing...

On some a posteriori error estimation results for the method of lines

Segeth, Karel, Šolín, Pavel

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The paper is an attempt to present an (incomplete) historical survey of some basic results of residual type estimation procedures from the beginning of their development through contemporary results to future prospects. Recently we witness a rapidly increasing use of the $h p$ -FEM which is due to the well-established theory. However, the conventional a posteriori error estimates (in the form of a single number per element) are not enough here, more complex estimates are needed, and this...

Anisotropic $h p$ -adaptive method based on interpolation error estimates in the $H^{1}$ -seminorm

Vít Dolejší (2015)

Applications of Mathematics

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We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken $H^{1}$ -seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of...

Uniform $L^{1}$ error bounds for semi-discrete finite element solutions of evolutionary integral equations

Lin, Qun, Xu, Da, Zhang, Shuhua

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In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem $u_{t} + \int_{0}^{t} β (t - s) A u (s) d s = 0, u (0) = v, t > 0,$ where A is an elliptic partial-differential operator and $β (t)$ is positive, nonincreasing and log-convex on $(0, \infty)$ with $0 \leq β (\infty) < β (0^{+}) \leq \infty$ . Error estimates are derived in the norm of $L_{t}^{1} (0, \infty; L_{x}^{2})$ , and some estimates for the first order time derivatives of the errors are also given.

Displaying similar documents to “New a posteriori L ∞ ( L 2 ) and L 2 ( L 2 ) -error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems”

Displaying similar documents to “New a posteriori $L^{\infty} (L^{2})$ and $L^{2} (L^{2})$ -error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems”