Advanced Search
Article Contents
Article Contents

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

  • * Corresponding author: Mariano Mateos

    * Corresponding author: Mariano Mateos 
The project was supported by DFG through the International Research Training Group IGDK 1754 Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures.
The second author was partially supported by the Spanish Ministerio Español de Economía y Competitividad under research projects MTM2014-57531-P and MTM2017-83185-P.
Abstract Full Text(HTML) Figure(6) Related Papers Cited by
  • The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasi-uniform meshes and superconvergent meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in nonconvex domains are provided.

    Mathematics Subject Classification: 65N30, 65N15, 49M05, 49M25.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 6.  Constrained problems. Experimental orders of convergence vs biggest angle. Left: generic case. Right: worst case.

    Figure 1.  Convergence rates depending on the maximal interior angle in the unconstrained case

    Figure 2.  Convergence rates depending on the maximal interior angle in the constrained case

    Figure 3.  Family of quasi-uniform meshes which is not $O(h^2)$-irregular

    Figure 4.  Family of quasi-uniform $O(h^2)$-irregular meshes

    Figure 5.  Unconstrained problems. Experimental orders of convergence vs biggest angle.

  • [1] T. ApelM. MateosJ. Pfefferer and A. Rösch, On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains, SIAM J. Control Optim., 53 (2015), 3620-3641.  doi: 10.1137/140994186.
    [2] T. ApelS. Nicaise and J. Pfefferer, Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains, Numer. Methods Partial Differential Equations, 32 (2016), 1433-1454.  doi: 10.1002/num.22057.
    [3] T. ApelJ. Pfefferer and A. Rösch, Finite element error estimates on the boundary with application to optimal control, Mathematics of Computation, 84 (2015), 33-70.  doi: 10.1090/S0025-5718-2014-02862-7.
    [4] C. BacutaJ. Bramble and J. Xu, Regularity estimates for elliptic boundary value problems in Besov spaces, Mathematics of Computation, 72 (2003), 1577-1595.  doi: 10.1090/S0025-5718-02-01502-8.
    [5] R. Bank and J. Xu, Asymptotically exact a posteriori error estimators, part Ⅰ: Grids with superconvergence, SIAM Journal on Numerical Analysis, 41 (2003), 2294-2312.  doi: 10.1137/S003614290139874X.
    [6] S. BartelsC. Carstensen and G. Dolzmann, Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis, Numerische Mathematik, 99 (2004), 1-24.  doi: 10.1007/s00211-004-0548-3.
    [7] M. Berggren, Approximations of very weak solutions to boundary-value problems, SIAM J. Numer. Anal., 42 (2004), 860-877 (electronic).  doi: 10.1137/S0036142903382048.
    [8] E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611 (electronic).  doi: 10.1137/050626600.
    [9] E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic control problems, in Analysis and Optimization of Differential Systems, Springer, 2003, 89-100.
    [10] P. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis (eds. P. Ciarlet and J. Lions), vol. Ⅱ. Finite Element Methods (Part 1), North-Holland, 1991, 17-352.
    [11] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043.
    [12] K. DeckelnickA. Günther and M. Hinze, Finite element approximation of Dirichlet boundary control for elliptic PDEs on two-and three-dimensional curved domains, SIAM J. Control Optim., 48 (2009), 2798-2819.  doi: 10.1137/080735369.
    [13] A. DemlowD. LeykekhmanA. Schatz and L. Wahlbin, Best approximation property in the w1 norm for finite element methods on graded meshes, Mathematics of Computation, 81 (2012), 743-764.  doi: 10.1090/S0025-5718-2011-02546-9.
    [14] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.
    [15] T. HorgerJ. Melenk and B. Wohlmuth, On optimal L2-and surface flux convergence in FEM, Computing and Visualization in Science, 16 (2013), 231-246.  doi: 10.1007/s00791-015-0237-z.
    [16] M. Mateos, Optimization methods for Dirichlet control problems, to appear in Optimization, https://arXiv.org/abs/1701.07619.
    [17] M. Mateos and I. Neitzel, Dirichlet control of elliptic state constrained problems, Comput. Optim. Appl., 63 (2016), 825-853.  doi: 10.1007/s10589-015-9784-y.
    [18] S. MayR. Rannacher and B. Vexler, Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems, SIAM Journal on Control and Optimization, 51 (2013), 2585-2611.  doi: 10.1137/080735734.
    [19] J. Melenk and B. Wohlmuth, Quasi-optimal approximation of surface based lagrange multipliers in finite element methods, SIAM Journal on Numerical Analysis, 50 (2012), 2064-2087.  doi: 10.1137/110832999.
    [20] S. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, vol. 13 of De Gruyter Expositions in Mathematics, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110848915.525.
    [21] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Corrected 2nd edition, Monographs and Studies in Mathematics, Springer Berlin Heidelberg, 2012.
    [22] J. Pfefferer, Numerical analysis for elliptic Neumann boundary control problems on polygonal domains, PhD Thesis, Universität der Bundeswehr München, 2014, http://athene.bibl.unibw-muenchen.de:8081/node?id=92055.
    [23] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Mathematics of Computation, 38 (1982), 437-445.  doi: 10.1090/S0025-5718-1982-0645661-4.
  • 加载中



Article Metrics

HTML views(2062) PDF downloads(293) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint