Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems
The purpose of this paper is the identification of a parameter in the elliptic equation \[ -\nabla \cdot (q(x)\nabla u(x))=f(x)\quad \text{in }\Omega \] or in the parabolic equation \[ u_t-\nabla \cdot (q(x)\nabla u)=g(x,t)\quad \text{in }\Omega\times (0,T), \] with Neumann homogeneous boundary conditions and an initial condition for the parabolic problem. The focus of this work is to find both the solution \(u\) and the distributed coefficient \(q\), when some measurement of \(u\) is given. By means of the Tikhonov's regularization, the inverse problem of reconstructing \(q\) may be formulated as a minimization problem. By using piecewise linear finite element approximations to the regularization systems, \(L^2\) and energy-norm error estimates for the approximations of the coefficient \(q\) are established.
scientific article; zbMATH DE number 5837953