Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems

Finite element methods for coefficient identification in an elliptic equation

“…This is different from that of Falk [19], of Kohn and Lowe [8], and of Wang and Zou [9]; however, it is unavoidable for numerical computations, since in general we cannot define the pointwise values of z δ ∈ H 1 ( ) at the nodes of (T h ). We will discuss also the functional without data mollification…”

“…When z δ ∈ H 1 ( ), the difference between the methods in [8,9,19] and that of ours is that these authors directly use the data z δ in their optimization problems while we use its mollified version h z δ , see (2.15). However, for finite element methods, in general we cannot define pointwise values of z δ at the nodes of the triangulation T h , if z δ ∈ H 1 ( ).…”

“…[4][5][6][7] In this paper, we apply the finite element method to this optimization problem and prove some convergence results and error estimates of the method. We note that several researchers, such as Falk [19], Kaltenbacher and Schöberl [14], Kohn and Lowe [8], Kärkkäinen [20], Zou [17], and Wang and Zou [9], used the finite element method to the above coefficient identification problem. However, each author used different methods for solving it and assumed different hypotheses on the solutions, and so they obtained different convergence results.…”

“…Concerning such a type of observations, we refer the reader to [7][8][9][10][11][12][14][15][16], where detailed discussions about the ill-posedness of the identification problem and interpolations of discrete measurements of the solution u resulting the data z δ satisfying (1.4) are given. For solving the above identification problem, in [4][5][6][7] we followed Knowles [16,18] and Zou [17] in minimizing the so-called energy functional…”

“…from aquifer analysis. For surveys on the subject, we refer the reader to [1][2][3][4][5][6][7][8][9] and the references therein.…”