Adaptive Methods for Parameter Identification in Ground Water Hydrology

The paper considers the simultaneous identification of parameters and source term appearing in elliptic Partial Differential Equations from knowledge of the reference potential, which satisfies Dirichlet boundary conditions. Identifying the unknowns means complying with the equation error criterion. The adaptive method consists of transforming both the potential and the unknown parameters into time-dependent quantities, which evolve according to given laws. In the continuum case, the time decay property of a Lyapunov function is shown to hold.

Next, the finite-dimensional (Galerkin) approximation of the evolution equations is considered and, given the existence of a solution, the pertaining convergence results are stated s. t., the time-dependent potential approaches the reference potential and the unknowns asymptotically comply with the time-independent equation.

Then, the discrete identification problem is stated. The finite element method is used to obtain the discrete equation error criterion. A Eulerian numerical scheme for all discretized evolution equations is provided. The sequences of the corresponding solutions i. e., the potential and the unknowns, are shown to converge. The detailed proof is provided.

Finally, the above method is applied to the identification of hydraulic conductivity, leakage parameter and source term of a two-layered aquifer in the Aachtal (Germany) area. The numerical results obtained from noiseless and noisy potential data are provided.