Finite element methods for coefficient identification in an elliptic equation

Abstract

We study the problem of identifying the spatially varying diffusion coefficient in the boundary value problems for the elliptic equation in , on and on , , , when the solution is imprecisely given by with and . The finite element method is applied to a convex energy functional with Tikhonov regularization for solving this coefficient identification problem. We show the -convergence of finite element solutions to the unique minimum norm solution of the identification problem. Furthermore, convergence rates of the method are established under certain source conditions.

Keywords:

• coefficient identification
• elliptic equations
• Neumann problem
• Dirichlet problem
• mixed boundary value problems
• ill-posed problems
• Tikhonov regularization
• finite element methods
• convergence rates
• error estimates

AMS Subject Classifications:

• 65N21
• 65N12
• 35J25
• 35R30

Acknowledgments

This research was supported by a “Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme”, by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.02-2011.50.