While the mimetic finite-difference method shares many similarities with the finite-element and finite-volume methods, it has the advantage of naturally accommodating grids with arbitrary polyhedral elements. In this study, we use this attribute to develop an adaptive scheme for the solution of the geophysical electromagnetic modelling problem on unstructured grids. Starting with an initial tetrahedral grid, our mesh adaptivity implements an iterative h-refinement where a residual- and jump-based goal-oriented error estimator is used to mark a certain portion of the elements. The marked elements are decomposed into new tetrahedra by regular subdivision, creating an octree-like unstructured grid. Since arbitrary polyhedra are naturally permitted in the mimetic finite-difference method, the added nodes are not regarded as hanging nodes and hence any level of non-conformity can be implemented without a modification to the mimetic scheme. In this study, we consider 2-irregularity where two levels of non-conformity between the adjacent elements is permitted. We use a total field approach where the electric field is defined at the edges of the polyhedral elements and the electromagnetic source may have an arbitrary shape and location. The accuracy of the mimetic scheme and the effectiveness of the proposed mesh adaptivity are verified using benchmark and realistic examples that represent various magnetotelluric and controlled-source survey scenarios. The mesh adaptivity generates grids with refinement generally concentrated at the transmitter and receiver locations and the interfaces of materials with contrasting conductivities, and the mimetic finite-difference solutions have good agreement with the reference numerical and real data. We also demonstrate the practicality of our method using an example with an analytical solution and comparison with a standard mesh regeneration technique. The results show that our mesh adaptivity procedure can result in a higher accuracy, with similar numbers of elements, when compared with the mesh regeneration approach. Also, using a generic sparse direct solver, our method is found to be more efficient than the mesh regeneration approach in terms of computation time and memory usage. Moreover, a comparison between 1- and 2-irregularity shows the higher efficiency of the latter in terms of the number of elements required to reach a certain level of accuracy.

尽管模拟有限差分法与有限元和有限体积法有许多相似之处，但它的优点是可以自然地容纳具有任意多面体元素的网格。在这项研究中，我们使用此属性来开发一种自适应方案，以解决非结构化网格上的地球物理电磁建模问题。从初始的四面体网格开始，我们的网格自适应实现了迭代h-优化，其中基于残差和跳跃的面向目标的误差估计器用于标记元素的特定部分。通过常规细分，标记的元素将分解为新的四面体，从而形成类似八叉树的非结构化网格。由于在模拟有限差分方法中自然允许使用任意多面体，因此添加的节点不被视为悬挂节点，因此无需修改模拟方案即可实现任何级别的不合格。在这项研究中，我们考虑了2-不规则性，其中相邻元素之间的两个级别的不符合被允许。我们使用全场方法，其中在多面体元素的边缘定义了电场，并且电磁源可以具有任意形状和位置。使用代表各种大地电磁和受控源调查方案的基准和实际示例，验证了模拟方案的准确性和所提出的网格自适应性的有效性。网格自适应性生成的网格细化后通常集中在发射器和接收器的位置，并且具有导电性相反的材料界面，并且模拟有限差分解决方案与参考数值和实际数据具有良好的一致性。我们还将通过一个示例和一个解析解决方案，并与一个标准的网格再生技术进行比较，来证明我们方法的实用性。结果表明，与网格再生方法相比，我们的网格自适应程序可以在具有相似数量元素的情况下实现更高的精度。还，使用通用的稀疏直接求解器，我们的方法在计算时间和内存使用方面比网格再生方法更有效。此外，在1和2不规则性之间进行比较显示，就达到一定精度水平所需的元素数量而言，后者的效率更高。