Cubic-Spline Multiscale Finite Element Method for Solving Nodal Darcian Velocities in Porous Media

Abstract

This paper presents a cubic-spline multiscale finite element method (CMSFEM) for solving groundwater flow problems in porous media. The main idea of this method is using the multiscale finite element method (MSFEM) to efficiently solve the hydraulic heads and nodal Darcian velocities. CMSFEM employs the cubic-spline technique to obtain continuous nodal head derivatives, which ensures the continuity of velocities. Furthermore, CMSFEM can not only solve velocities at coarse-scale grid nodes but also solve those at the fine-scale nodes in each coarse element grid. Instead of solving the full study region problem, the computation of the fine-scale velocities is decoupled from coarse element to coarse element, which can be implemented in parallel. Therefore, CMSFEM saves much computational costs in solving heads and velocities, which is important for high computation problems. The applications in this paper demonstrate that the CMSFEM has high accuracy and efficiency in solving velocities and heads.