Mass Lumping for the Optimal Control of Elliptic Partial Differential Equations

The finite element discretization of a control constrained elliptic optimal control problem is studied. Control and state are discretized by higher order finite elements. The inequality constraints are only posed in the Lagrange points. The computational effort is significantly reduced by a new mass lumping strategy. The main contribution is the derivation of new a priori error estimates up to order $h^4$ on locally refined meshes. Moreover, we propose a new algorithmic strategy to obtain such highly accurate results. The theoretical findings are illustrated by numerical examples.

[1]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[2]  G. Fix Review: Philippe G. Ciarlet, The finite element method for elliptic problems , 1979 .

[3]  Andreas Springer,et al.  Third order convergent time discretization for parabolic optimal control problems with control constraints , 2013, Computational Optimization and Applications.

[4]  René Schneider,et al.  A Posteriori Error Estimation for Control-Constrained, Linear-Quadratic Optimal Control Problems , 2016, SIAM J. Numer. Anal..

[5]  Winfried Sickel,et al.  Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations , 1996, de Gruyter series in nonlinear analysis and applications.

[6]  Konstantin Pieper,et al.  Finite element discretization and efficient numerical solution of elliptic and parabolic sparse control problems , 2015 .

[7]  Eduardo Casas,et al.  Error estimates for the numerical approximation of Neumann control problems , 2008, Comput. Optim. Appl..

[8]  Arnd Rösch,et al.  Error estimates for linear-quadratic control problems with control constraints , 2006, Optim. Methods Softw..

[9]  René Schneider,et al.  Achieving optimal convergence order for FEM in control constrained optimal control problems , 2015 .

[10]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[11]  Michael Hinze,et al.  A note on the approximation of elliptic control problems with bang-bang controls , 2010, Computational Optimization and Applications.

[12]  Gerd Wachsmuth,et al.  Convergence and regularization results for optimal control problems with sparsity functional , 2011 .

[13]  T. Geveci,et al.  On the approximation of the solution of an optimal control problem governed by an elliptic equation , 1979 .

[14]  Thomas Apel Interpolation in h‐Version Finite Element Spaces , 2004 .

[15]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[16]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

[17]  Jean E. Roberts,et al.  Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation , 2000, SIAM J. Numer. Anal..

[18]  T. Apel Anisotropic Finite Elements: Local Estimates and Applications , 1999 .

[19]  Roland Herzog,et al.  Approximation of sparse controls in semilinear equations by piecewise linear functions , 2012, Numerische Mathematik.

[20]  M. Rivara,et al.  Local modification of meshes for adaptive and/or multigrid finite-element methods , 1991 .

[21]  Djalil Kateb,et al.  On the boundedness of the mapping f ↦ |f|μ, μ > 1 on Besov spaces , 2003 .

[22]  Richard S. Falk,et al.  Approximation of a class of optimal control problems with order of convergence estimates , 1973 .

[23]  Arnd Rösch,et al.  Finite element error estimates for Neumann boundary control problems on graded meshes , 2011, Computational Optimization and Applications.

[24]  Anders Logg,et al.  The FEniCS Project Version 1.5 , 2015 .

[25]  Arnd Rösch,et al.  Superconvergence Properties of Optimal Control Problems , 2004, SIAM J. Control. Optim..

[26]  David Sevilla,et al.  Polynomial integration on regions defined by a triangle and a conic , 2010, ISSAC.

[27]  Y. Meyer,et al.  Fonctions qui opèrent sur les espaces de Sobolev , 1991 .

[28]  Arnd Rösch,et al.  Error estimates for finite element approximations of elliptic control problems , 2007 .