Different types of grids in geometry12/30/2023 ![]() Much of this is subject to current petrophysical research, which gives us new insights into the fundamental properties of these parameters (e.g., Börner et al. Moreover, texture or heterogeneity below a certain spatial scale can appear as anisotropy, leading to a matrix representation of electromagnetic parameters in the equations to be discretized (Weidelt 1999 Li and Pek 2008 Yan et al. Electric conductivity can become complex-valued, frequency dependent and nonlinear. ![]() 2020) where a variety of pore-scale and inner-surface electrochemical processes are usually not covered anymore by our chosen mathematical model. On a small scale, we may end up in microscopic, possibly self-similar material structures (Pape et al. Particularly, the geometry of our physical models in the Earth sciences is extremely challenging due to its multi-scale nature. They therefore contain a multitude of sources of error and are subject to a limited range of validity. As a matter of fact, these models are always an abstract and incomplete representation of this reality. The goal of electromagnetic (EM) modeling in geophysics is to find mathematical and physical models that represent as well as possible the reality of our perceived geo-environment and allow us to predict and understand the behavior of EM fields in space and time. It will be shown that the sensitivity is the most important function in both guiding the geometric mapping and the local refinement. Therefore, an overview of the most common a posteriori error estimators is given. Although the available error estimators do not necessarily provide reliable error bounds for our complex geomodels, they are still useful to guide grid refinement. ![]() In summary, the accuracy of the solution depends on the geometric mapping, the choice of the mathematical model, and the spatial discretization. After a brief outline of early methods and modeling approaches, the paper mainly discusses the capabilities of the finite element method formulated on unstructured grids and the advantages of local h-refinement allowing for both a flexible and largely accurate representation of the geometries of the multi-scale geomaterial and an accurate evaluation of the underlying functions representing the physical fields. It discusses ways of estimating errors of our solutions for a perfectly matched modeling domain and the problems that arise from its insufficient representation. This review paper addresses the development of numerical modeling of electromagnetic fields in geophysics with a focus on recent finite element simulation.
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