Electromagnetic Modeling Using Adaptive Grids - Error Estimation and Geometry Representation

This review paper addresses the development of numerical modeling of electromagnetic fields in geophysics with a focus on recent finite element simulation. It discusses ways of estimating errors of our solutions for a perfectly matched modeling domain and the problems that arise from its insufficient representation. 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. In summary, the accuracy of the solution depends on the geometric mapping, the choice of the mathematical model, and the spatial discretization. Although the available error estimators do not necessarily provide reliable error bounds for our complex geomodels, they are still useful to guide grid refinement. Therefore, an overview of the most common a posteriori error estimators is given. It will be shown that the sensitivity is the most important function in both guiding the geometric mapping and the local refinement.