3-D forward modelling for DC resistivity method based on smooth multiscale finite-element algorithm

When the conventional finite-element method is used to simulate the 3-D direct-current (DC) resistivity response over a conductive earth with large complex structures, it requires finely discretized mesh to accurately represent the underground structures. Directly solving the current-conduction problem on the fine mesh will lead to a huge amount of calculation. In this paper, we develop a fast 3-D forward modelling method for DC resistivity method based on smooth multiscale finite-element algorithm. Instead of using the conventional polynomial basis functions, we construct the multiscale basis functions by solving the local boundary-value problems of partial differential equation in parallel on the multiscale meshes. The multiscale basis functions can capture the small-scale heterogeneous information in coarse cells and reflect it to the large scale by assembling macro matrix of coarse mesh. Thus, it enables us to quickly obtain the accurate solution by solving the original current-conduction problem with complex structures on coarse mesh. We further adopt the oversampling technology to improve the forward modelling accuracy. Besides, by combining with the gradient smoothing technology, we avoid establishing the continuous form of multiscale basis functions and their spatial derivative integral operation to rapidly assemble the macro matrix. Finally, the reliability of the proposed algorithm is verified by applying our code to the 3-D complex models and comparing it with the conventional finite-element method.