An analysis of groundwater flow in a finite region with a sinusoidal top

In this paper we consider a groundwater flow problem in a semi-infinite vertical region bounded on top by a sloping sinusoidal curve dealt with by Shivakumar et al. (Shivakumar, P.N.; Williams, JJ.; Qiang Ye.; Chuanxiang, Ji. An Analysis of Groundwater Flow in an Infinite Region with Sinusoidal Top. Numer. Funct. Anal. and Optimiz. 2000, 21(l&2), 263-271). We first extend the analysis to the more realistic case of a finite region Omega. The problem is described by an elliptic equation for the hydraulic head under given boundary conditions. The obtained results are in good agreement with a classical result (Toth, J. A Theoretical Analysis of Groundwater Flow in Small Drainage Basins. J. Geophys. Res. 1963, 67, 4795-4812). We extend the problem further by allowing Dirichlet-type boundary conditions on the lower boundary. Our first goal is to obtain a semi-analytical solution. The problem is reduced to an infinite system of linear equations by using a suitable Fourier expansion method, transforming lower edge boundary conditions to a cosine series and by construction of a Gramm matrix. The error estimate from Shivakumar et al. is used to get the best match solution on the top boundary. Some computational results are presented, as well as the results obtained from applying a standard Galerkin finite element method and an infinite element method to the problem. This allows deduction of the cases in which one method prevails over the other.