Partition of unity methods for approximation of point water sources in porous media

Several partition of unity methods (PUM) are compared on the problem of steady water flow in an aquifer-well system. In order to improve the approximation of a singular behavior of the pressure near the wells, the standard finite element space is enriched with a cut-off fundamental solution to a Laplace problem with a point source on the whole R2 space. The optimal order of convergence of PUM in terms of L2 norm of the error is demonstrated. The error of adaptive integration is analysed and a new adaptive strategy is proposed. The influence of the choice of the enriched domain is investigated and its impact on the error is demonstrated numerically.