Cubic-spline multiscale finite element method for simulation of nodal Darcy velocities in aquifers

A cubic-spline multiscale finite element method (MSFEM-C) is proposed for the simulation of nodal Darcy velocities in the heterogeneous media. The main goal of this method is to efficiently solve the hydraulic heads and nodal Darcy velocities. It is realized by the combination of cubic-spline technique and the multiscale finite element method (MSFEM). MSFEM-C applies cubic-spline technique to multiscale base functions so as to make their derivatives continuous. Therefore, the continuous derivatives of hydraulic head can be obtained, which ensures the continuity of the velocity field. The MSFEM-C is based on MSFEM, so that the computation of nodal Darcy velocities is decomposed from element to element. Therefore, MSFEM-C can save much computational cost, which makes it more efficinent in solving high computational problems, such as large-scale, long-term or nonlinear problems. The numerical experiments indicate that the MSFEM-C achieves accurate nodal Darcy velocities and hydraulic heads with much less computational cost.