Graded Meshes in Optimal Control for Elliptic Partial Differential Equations: An Overview
作者： Thomas Apel , Johannes Pfefferer , Arnd Rösch
DOI: 10.1007/978-3-319-05083-6_18
关键词: Polygon mesh 、 Elliptic partial differential equation 、 Mathematics 、 Applied mathematics 、 Convergence (routing) 、 Domain (mathematical analysis) 、 Finite element method 、 Optimal control 、 Numerical partial differential equations 、 Boundary value problem
摘要: It is well known that singularities in the solution of boundary value problems due to corners and edges domain lead a reduction convergence order standard finite element method when quasi-uniform meshes are used. also locally graded suited recover optimal order. Less critical angles mesh grading becomes necessary; it not always same but depends on norm which error estimated. In this paper, an overview results given lacking estimates pointed out. Since for control based those pure both cases considered.
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sci-hub.st 参考文章(25)
Tunc Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique. ,vol. 13, pp. 313- 328 ,(1979) , 10.1051/M2AN/1979130403131
C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints Control and Cybernetics. ,vol. 37, pp. 51- 83 ,(2008)
Anna-Margarete Sändig, Error estimates for finite-element solutions of elliptic boundary value problems in non-smooth domains Zeitschrift Fur Analysis Und Ihre Anwendungen. ,vol. 9, pp. 133- 153 ,(1990) , 10.4171/ZAA/388
T. Apel, M. Dobrowolski, Anisotropic interpolation with applications to the finite element method Computing. ,vol. 47, pp. 277- 293 ,(1991) , 10.1007/BF02320197
Thomas Apel, Johannes Pfefferer, Max Winkler, Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra Mathematical Methods in The Applied Sciences. ,vol. 39, pp. 1206- 1232 ,(2016) , 10.1002/MMA.3566
Thomas Apel, Gunter Winkler, Optimal control under reduced regularity Applied Numerical Mathematics. ,vol. 59, pp. 2050- 2064 ,(2009) , 10.1016/J.APNUM.2008.12.003
M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case Computational Optimization and Applications. ,vol. 30, pp. 45- 61 ,(2005) , 10.1007/S10589-005-4559-5
L.A. Oganesyan, L.A. Rukhovets, Variational-difference schemes for linear second-order elliptic equations in a two-dimensional region with piecewise smooth boundary USSR Computational Mathematics and Mathematical Physics. ,vol. 8, pp. 129- 152 ,(1968) , 10.1016/0041-5553(68)90008-6
Richard S. Falk, Approximation of a class of optimal control problems with order of convergence estimates Journal of Mathematical Analysis and Applications. ,vol. 44, pp. 28- 47 ,(1973) , 10.1016/0022-247X(73)90022-X
Thomas Apel, Arnd Rösch, Dieter Sirch, $L^\infty$-Error Estimates on Graded Meshes with Application to Optimal Control Siam Journal on Control and Optimization. ,vol. 48, pp. 1771- 1796 ,(2009) , 10.1137/080731724