Adaptive Finite Element Numerical Simulation, Based On The Dc Resistivity Of Unstructured Grid

Currently, the direct current resistivity models are simulated by the typical or one-time finite-element method (TFEM or OFEM), which often associated with the structured mesh such as hexahedron or pentahedral in 3D or with simple unstructured by directly dividing unstructured mesh in 3D or 2D. Obviously there is a serious problem that when complex geometrical models are considered the numerical solutions become unreliable. The first reason is that once the resistivity model is meshed the accuracy of one-time finite-element process is also determined and can not be updated. Generally the analytical expressions of most of complex models are unknown. If we want to achieve reliable numerical solutions, based on convergence rate of one-time finite-element method, we must decrease the size of finite-element as small as we can and simultaneously adopt high order finite-element (such as quadric, cubic even more high order). Unfortunately this should lead to huge numbers of nodes (or degrees of freedom, Dofs) and computational cost which make the finite-element numerical simulation on personal computers (PCs) impractical. The second reason is that whatever regular or irregular structured meshes are adopted it is very difficult for them to conform the complex curve or surfaces boundaries. This inconsistent approximation should lead to large discrete errors in numerical solutions.In this study, we present a new adaptive finite element method associating with unstructured mesh. By the unstructured mesh, the geometrical error caused by unsuitable discretization can be removal significantly. And the iterative process with adaptively and locally refined meshes can make our numerical solutions convergence to the true solutions which are even unknown at all. In each refinement, our iterative mesh is nearly optimal therefore we can confirm that the whole iterative process will be terminated after several steps. And also with the quality elements in each iterative mesh, the density of nodes and elements in sharp gradient can significantly decrease our computational cost while it also keeps the accurate numerical solutions. The unstructured mesh is generated by the Delaunay refinement algorithm with high quality guarantee, which will significantly decrease its bad effects in the final numerical solutions. As the adaptive strategies, the h-version adaptive way instead of p-version or hp-version ones is adopted as its simplest implementation with also high convergence rate. In the key role of h-version adaptive process, a-posterior error estimator proposed by Zienkiewicz and Zhu, which is well-known as Z-Z error estimator, is also adopted to drive our adaptive mesh refinement process. This error estimator technique is based on the gradient recovery technique specially the superconvergence patch recovery method. Numerous examples in this paper show that our adaptive iterations can be terminated in less than 4-5 steps with less than 0.5%-1.0% relative error in finial numerical solutions(apparent resistivities), and also the singularity effects caused by the current source points can be also efficiently removal.Therefore, based on the technique proposed in this study, we can efficiently solve the problem both the accuracy of numerical solutions and the computational cost such as run-time and memory size. And also we believe that our technique could be more and more widely applied in the resistivity explorations even in the whole geophysical electromagnetic exploration fields just on personal computers(PCs).