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.

中文翻译：

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基于hp自适应有限元法的三维直流电阻率正演建模

摘要 近年来，自适应有限元方法已被用于直流电阻率正演建模，以提高数值解的精度。自适应有限元解的精度主要受两个因素影响：单元大小（h）和形状函数的阶数（p）。为了进一步提高自适应有限元解的精度并保持较低的计算成本，我们提出了一种结合 h 自适应和 p 自适应的 hp 自适应有限元算法，用于三维直流正向建模。自适应网格细化由解的后验误差估计量和平滑度估计量指导。在自适应过程中，对于误差较大的单元格，如果解的平滑度低，则该单元格会被细化；除此以外，单元格上的形状函数的阶数将增加一个阶数。为了获得具有地形复杂模型的高精度有限元解决方案，采用了非结构化网格。采用基于八叉树的网格细化方法对单元进行细化，利用一维多项式的张量积生成三维空间中任意阶数的形状函数。使用两层模型来验证算法的正确性。全局、h 自适应和 hp 自适应细化算法的收敛速度的比较表明，我们的算法可以以最低的计算成本提供最准确的解决方案。它显示了指数收敛速度。最后，