Convergence and quasi-optimality of an adaptive finite element method for elliptic Robin boundary control problem

Abstract In this paper we focus on the convergence and quasi-optimality of an adaptive finite element method for elliptic Robin boundary control problems. We use piecewise linear finite elements to approximate the state and the adjoint state variables, and the variational discretization to approximate the control variable. Under mild assumption on the initial mesh, we prove the contraction property, for the sum of the energy errors of the state and adjoint state and the scaled error estimator, on two consecutive adaptive loops. The resulting linear convergence yields the quasi-optimal convergence rate for the AFEM algorithm applied to our problem. Additionally, some numerical results are provided to support our theoretical analysis.

[1]  Wenbin Liu,et al.  A Posteriori Error Estimates for Distributed Convex Optimal Control Problems , 2001, Adv. Comput. Math..

[2]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs , 2005, SIAM J. Numer. Anal..

[3]  Shipeng Mao,et al.  Quasi-optimality of an Adaptive Finite Element Method for an Optimal Control Problem , 2011, Comput. Methods Appl. Math..

[4]  Christian Kreuzer,et al.  Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method , 2008, SIAM J. Numer. Anal..

[5]  Jean-Pierre Raymond,et al.  ESTIMATES FOR THE NUMERICAL APPROXIMATION OF DIRICHLET BOUNDARY CONTROL FOR SEMILINEAR ELLIPTIC EQUATIONS , 2006 .

[6]  Ricardo H. Nochetto,et al.  Data Oscillation and Convergence of Adaptive FEM , 2000, SIAM J. Numer. Anal..

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

[8]  Faker Ben Belgacem,et al.  Singular perturbation for the Dirichlet boundary control of elliptic problems , 2003 .

[9]  Rob P. Stevenson,et al.  An Optimal Adaptive Finite Element Method , 2004, SIAM J. Numer. Anal..

[10]  ROB STEVENSON,et al.  The completion of locally refined simplicial partitions created by bisection , 2008, Math. Comput..

[11]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[12]  S. Agmon.  Lectures on Elliptic Boundary Value Problems , 1965 .

[13]  Rob P. Stevenson,et al.  Optimality of a Standard Adaptive Finite Element Method , 2007, Found. Comput. Math..

[14]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[15]  Rolf Rannacher,et al.  Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept , 2000, SIAM J. Control. Optim..

[16]  Wenbin Liu,et al.  Local A Posteriori Error Estimates for Convex Boundary Control Problems , 2009, SIAM J. Numer. Anal..

[17]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[18]  Ronald H. W. Hoppe,et al.  Convergence Analysis of an Adaptive Finite Element Method for Distributed Control Problems with Control Constraints , 2007 .

[19]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[20]  Konstantinos Chrysafinos,et al.  Error Estimates for Discontinuous Galerkin Time-Stepping Schemes for Robin Boundary Control Problems Constrained to Parabolic PDEs , 2014, SIAM J. Numer. Anal..

[21]  Wenbin Liu,et al.  A Posteriori Error Estimates for Control Problems Governed by Stokes Equations , 2002, SIAM J. Numer. Anal..

[22]  Jean-Pierre Raymond,et al.  Penalization of Dirichlet optimal control problems , 2009 .

[23]  Kunibert G. Siebert,et al.  Convergence of Adaptive Finite Elements for Optimal Control Problems with Control Constraints , 2014 .

[24]  Wei Gong,et al.  Adaptive finite element method for elliptic optimal control problems: convergence and optimality , 2015, Numerische Mathematik.

[25]  Yanping Chen,et al.  Convergence and quasi-optimality of an adaptive finite element method for optimal control problems with integral control constraint , 2018, Adv. Comput. Math..

[26]  Kunibert G. Siebert,et al.  A Posteriori Error Analysis of Optimal Control Problems with Control Constraints , 2014, SIAM J. Control. Optim..

[27]  Zhaojie Zhou.  A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems , 2016, Int. J. Comput. Math..

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

[29]  Zhaojie Zhou,et al.  Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints , 2014 .

[30]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems , 2005, Comput. Optim. Appl..

[31]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

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

[33]  Arnd Rösch,et al.  On saturation effects in the Neumann boundary control of elliptic optimal control problems , 2007, Comput. Optim. Appl..

[34]  Jia Feng,et al.  An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems , 2004, Math. Comput..

[35]  S. Ravindran,et al.  A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier--Stokes Equations , 1998 .

[36]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

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