Multiscale finite element-finite element model for simulating nodal Darcy velocity

A multi-scale finite element-finite element model (MSFEM-FEM) is proposed, and it can effectively simulate the nodal Darcy velocity and ensure the velocity continuity. The MSFEM-FEM employs the multi-scale finite element method (MSFEM) to replace the head simulation part of the Yeh's finite element model, thus to improve the efficiency and accuracy. Then, the MSFEM-FEM transforms the multi-scale grid to the finite element one, thus, it can directly apply the Yeh's finite element model to obtain continuous Darcy velocity. Based on the multi-scale basis function, the MSFEM-FEM can extract the global information of the study area which allows it to obtain the accurate head solution efficiently on the coarse scale. By transforming the coarse-scale grid into the finite element grid, the MSFEM-FEM can directly employ the Yeh's finite element model to import the global information from the head solution into the Darcy velocity, which can also improve the accuracy of Darcy velocity and ensure the velocity continuity. In addition, the MSFEM-FEM can apply multi-scale basis function to reconstruct the solutions, so as to obtain the fine-scale head and velocity solutions in the study area. The simulated results of two-dimensional groundwater problems show that the MSFEM-FEM can efficiently and accurately solve the head, Darcy velocity and flux, which outperforms the MSFEM and the Yeh's finite element model.