A Priori Error Estimates for the Finite Element Discretization of Elliptic Parameter Identification Problems with Pointwise Measurements

We develop an a priori error analysis for the finite element Galerkin discretization of parameter identification problems. The state equation is given by an elliptic partial differential equation of second order with a finite number of unknown parameters, which are estimated using pointwise measurements of the state variable.

[1]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[2]  Richard S. Falk,et al.  Error estimates for the numerical identification of a variable coefficient , 1983 .

[3]  Xuecheng Tai,et al.  Error Estimates for Numerical Identification of Distributed Parameters , 1992 .

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

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

[6]  A. H. Schatz,et al.  Interior estimates for Ritz-Galerkin methods , 1974 .

[7]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[8]  Roland Becker,et al.  A posteriori error estimation for finite element discretization of parameter identification problems , 2004, Numerische Mathematik.

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

[10]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem , 2002, Comput. Optim. Appl..

[11]  C. Kravaris,et al.  Identification of parameters in distributed parameter systems by regularization , 1983, The 22nd IEEE Conference on Decision and Control.

[12]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

[13]  William G. Litvinov,et al.  Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics , 2000 .

[14]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

[15]  A. H. Schatz,et al.  Maximum norm estimates in the finite element method on plane polygonal domains. I , 1978 .

[16]  R. Becker,et al.  Numerical parameter estimation for chemical models in multidimensional reactive flows , 2004 .

[17]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[18]  Tommi Kärkkäinen.  Error Estimates for Distributed Parameter Identification in Linear Elliptic Equations , 1996 .

[19]  Rolf Rannacher,et al.  Finite element methods for nonlinear elliptic systems of second order , 1980 .

[20]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[21]  Michael Hinze,et al.  Error Estimates in Space and Time for Tracking-type Control of the Instationary Stokes System , 2003 .