Three-dimensional direct current resistivity forward modeling based on the hp-adaptive finite element method

Abstract

In recent years, the adaptive finite element method has been used for direct current resistivity forward modeling to improve the accuracy of numerical solutions. The accuracy of adaptive finite element solutions is mainly affected by two factors: the cell size (h) and the order of the shape function (p). To further improve the accuracy of the adaptive finite element solutions and keep low computational costs, we present a hp-adaptive finite element algorithm, combining h-adaptive and p-adaptive, for three-dimensional direct current forward modeling. The adaptive mesh refinement is guided by a posteriori error estimator and a smoothness estimator of the solution. In the adaptive process, for a cell with a large error, if the smoothness of the solution is low, the cell will be refined; otherwise, the order of the shape function on the cell will be increased by one order. To obtain high-precision finite element solutions for complex models with topographies the unstructured grids are adopted. The octree-based mesh refinement method is used to refine the cells, and the shape functions with arbitrary orders in three-dimensional space are generated by using tensor products of one-dimensional polynomials. A two-layer model is used to verify the correctness of the algorithm. The comparison between the convergence rate of global, h-adaptive, and hp-adaptive refinement algorithms indicates that our algorithm can provide the most accurate solution with the lowest computation costs. It shows exponentially convergence rate. Finally, a topography model with an abnormal body and a complex abnormal model are used to verify the robustness of our algorithm.