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1
Content available Pseudospectral method for semilinear partial functional differential equations
100%
Czernous W.
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tom Vol. 30, no. 2
133-145
EN
We present a convergence result for two spectral methods applied to an initial boundary value problem with functional dependence of Volterra type. Explicit condition of Courant-Friedrichs-Levy type is assumed on time step τ and the number N of collocation points. Stability statements and error estimates are written using continuous norms in weighted Jacobi spaces.
2
Content available remote Convergence results for unbounded solutions of first order non-linear differential-functional equations
100%
Leszczyński H.
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nr 1
1-16
EN
We consider the Cauchy problem in an unbounded region for equations of the type either $D_{t}z(t,x) = f(t,x,z(t,x),z_{(t,x)},D_{x}z(t,x))$ or $D_{t}z(t,x)= f(t,x,z(t,x),z,D_{x}z(t,x))$. We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.
3
Content available remote Numerical approximation of first order partial differential equations with deviated variables
100%
Baranowska A. , Kamont Z.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2003
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tom [Z] 43(1)
1-31
EN
Classical solutions of the local Cauchy problem on the Haar pyramid are approximated in the paper by solutions of suitable quasilinear systems of difference equations. The proof of the stability of the difference problem is based on a comparison technique with nonlinear estimates of the Perron type. This new approach to the numerical solving of nonlinear equations with deviated variables is generated by a quasilinearization method for initial problems. Numerical examples are given.
4
Content available remote A note on variational discretization of elliptic Neumann boundary control
100%
Hinze M. , Matthes U.
Control and Cybernetics
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2009
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tom Vol. 38, no 3
577-591
EN
We consider variational discretization of Neumann-type elliptic optimal control problems with constraints on the control. In this approach the cost functional is approximated by a sequence of functionals, which are obtained by discretizing the state equation with the help of linear finite elements. The control variable is not discretized. Error bounds for control and state are obtained both in two and three space dimensions. Finally, we discuss some implementation issues of a generalized Newton method applied to the numerical solution of the problem class under consideration.
5
Content available Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems
100%
Merino P. , Neitzel I. , Tröltzsch F.
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tom 30
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nr 2
221-236
EN
In this paper we derive a priori error estimates for linear-quadratic elliptic optimal control problems with finite dimensional control space and state constraints in the whole domain, which can be written as semi-infinite optimization problems. Numerical experiments are conducted to ilustrate our theory.
6
Content available Error estimates for finite element approximations of elliptic control problems
100%
Alt W. , Bräutigam N. , Rösch A.
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tom 27
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nr 1
7-22
EN
We investigate finite element approximations of one-dimensional elliptic control problems. For semidiscretizations and full discretizations with piecewise constant controls we derive error estimates in the maximum norm.
7
Content available Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems
100%
Kahlbacher M. , Volkwein S.
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tom 27
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nr 1
95-117
EN
Proper orthogonal decomposition (POD) is a powerful technique for model reduction of linear and non-linear systems. It is based on a Galerkin type discretization with basis elements created from the system itself. In this work, error estimates for Galerkin POD methods for linear elliptic, parameter-dependent systems are proved. The resulting error bounds depend on the number of POD basis functions and on the parameter grid that is used to generate the snapshots and to compute the POD basis. The error estimates also hold for semi-linear elliptic problems with monotone nonlinearity. Numerical examples are included.
8
Content available Difference functional inequalities and applications
75%
Szafrańska A.
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2014
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tom Vol. 34, no. 2
405--423
EN
The paper deals with the difference inequalities generated by initial boundary value problems for hyperbolic nonlinear differential functional systems. We apply this result to investigate the stability of constructed difference schemes. The proof of the convergence of the difference method is based on the comparison technique, and the result for difference functional inequalities is used. Numerical examples are presented.
9
Content available remote Analysis and numerical approximation of an elastic frictional contact problem with normal compliance
75%
Han W. , Sofonea M.
Applicationes Mathematicae
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1999
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tom 26
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nr 4
415-435
EN
We consider the problem of frictional contact between an elastic body and an obstacle. The elastic constitutive law is assumed to be nonlinear. The contact is modeled with normal compliance and the associated version of Coulomb's law of dry friction. We present two alternative yet equivalent weak formulations of the problem, and establish existence and uniqueness results for both formulations using arguments of elliptic variational inequalities and fixed point theory. Moreover, we show the continuous dependence of the solution on the contact conditions. We also study the finite element approximations of the problem and derive error estimates. Finally, we introduce an iterative method to solve the resulting finite element system.
10
Content available remote A new stopping criterion for iterative solvers for control optimal problems
75%
Krumbiegel K. , Rosch A.
Archives of Control Sciences
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2008
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tom Vol. 18, no. 1
17-42
EN
Linear quadratic optimal control problems governed by PDEs with pointwise control constraints are considered. We derive error estimates for feasible and infeasible controls of the problem. Based on this theory an error estimator is constructed for different discretization schemes. Moreo ver, we establish the estimator as a stopping criterion for several optimization methods. Furthermore, additional errors caused by solving the linear systems are discussed. The theory is illustrated by numerical examples.
11
Content available remote A direct and accurate adaptive semi-Lagrangian scheme for the Vlasov-Poisson equation
75%
Pinto M. C.
International Journal of Applied Mathematics and Computer Science
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2007
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tom 17
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nr 3
351-359
EN
This article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1+1)-dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be said optimal in the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge in L^∞ towards the exact ones as ε and Δt tend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.
12
Content available Error estimates for the finite-element approximation of a semilinear elliptic control problem
63%
Casas E. , Troeltzsch F.
Control and Cybernetics
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2002
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tom Vol. 31, no 3
695-712
EN
We consider the finite-element approximation of a distributed optimal control problem governed by a semilinear elliptic partial differential equation, where pointwise constraints on the control are given. We prove the existence of local approximate solutions converging to a given local reference solution. Moreover, we derive error estimates for local solutions in the maximum norm.
13
Content available remote Convergence estimates for the acoustic scattering problem approximated by NURBS
63%
Karafiat A.
Computer Assisted Methods in Engineering and Science
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2013
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tom Vol. 20, no. 4
289--307
EN
The paper contains some estimates of an approximation to the solution of the problem of acoustic waves’s scattering by an elastic obstacle in two dimensions. The problem is approximated by the isogeometric adaptive method based on the known NURBS functions. The estimates show how the error of an approximation depends on the size of intervals and the degree of functions.
14
Content available remote Weighted difference schemes for systems of quasilinear first order partial functional differential equations
51%
Szafrańska A.
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tom Vol. 43, No. 2
225--251
EN
The paper deals with initial boundary value problems of the Dirichlet type for system of quasilinear functional differential equations. We investigate weighted difference methods for these problems. A complete convergence analysis of the considered difference methods is given. Nonlinear estimates of the Perron type with respect to functional variables for given functions are assumed. The proof of the stability of difference problems is based on a comparison technique. The results obtained here can be applied to differential integral problems and differential equations with deviated variables. Numerical examples are presented.
PL
Praca dotyczy zagadnień początkowo brzegowych typu Dirichlet’a dla układów quasiliniowych równań różniczkowo-funkcyjnych. Zamieszczona jest konstrukcja ważonych metod różnicowych dla wyjściowych zagadnień różniczkowych oraz przeprowadzona jest pełna analiza zbieżności. Niezbędne założenia obejmują oszacowania typu Perrona dla funkcji danych względem argumentów funkcyjnych. Dowód stabilności metody różnicowej opiera się na technice porównawczej. Teoretyczne rezultaty zobrazowane są na przykładzie całkowego równania różniczkowego oraz równań różniczkowych z odchylonym argumentem.
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