An inverse problem from the 2D-groundwater modelling

The paper is devoted to the inverse problem of identifying the coefficient in the main term of a quasilinear elliptic differential equation describing the filtration of groundwater. Experience suggests that the gradient of the piezometric head, i.e., Darcy's velocity, may have discontinuities and the transmissivity coefficient is a piecewise constant function. For solving this problem we use a modification of a direct method of G. Vainikko. Starting with a weak formulation of the problem a suitable discretization is obtained by the method of minimal error. If necessary this method can be combined with Tikhonov regularization. The main difficulty consists in generating distributed state observations from measurements of the ground-water level. For this step we propose an optimized data preparation procedure using additional information such as knowledge of the sought parameter values at some points and lower and upper bounds for the parameter. Numerical tests show that locally sufficiently many measurements provide locally satisfactory results. Two numerical examples, one with simulated data and the other with real life data, are given