SUMMARY

In this paper, we propose a spectral element method (SEM) based on unstructured tetrahedral grids for direct current (dc) resistivity modelling. Unlike the tensor-product of 1-D Gauss–Lobatto–Legendre (GLL) quadrature in conventional SEM, we use Proriol–Koornwinder–Dubiner (PKD) polynomials to form the high-order basis polynomials on tetrahedral grids. The final basis functions are established by using Vandermonde matrix. Compared to traditional SEM, our method truly takes into account the high precision of spectral method and the flexibility of finite element method with unstructured grids for modelling the complex underground structures. After addressing the theory on the construction of basis functions and interpolation and integration nodes, we validate our algorithm using the analytical solutions for a layered earth model and the results from other methods for multiple geoelectrical models. We further investigate a dual-track scheme for improving the accuracy of our SEM by increasing the order of interpolation polynomials or by refining the grids.

中文翻译：

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非结构四面体网格的基于谱元法的3-D直流电阻率建模

概要

在本文中，我们提出了一种基于非结构化四面体网格的谱元方法（SEM），用于直流（dc）电阻率建模。与传统SEM中一维高斯-洛巴托-勒根德（GLL）正交的张量积不同，我们使用Proriol-Koornwinder-Dubiner（PKD）多项式在四面体网格上形成高阶基多项式。最终的基础函数是通过使用Vandermonde矩阵建立的。与传统的SEM相比，我们的方法真正考虑了光谱法的高精度和非结构网格有限元方法的灵活性，可以对复杂的地下结构进行建模。在讨论了有关基函数以及插值和积分节点构造的理论之后，我们使用分层地球模型的解析解以及其他方法的多重地电模型的结果验证了我们的算法。我们进一步研究了通过增加插值多项式的阶数或完善网格来提高SEM精度的双轨方案。