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We present techniques for the efficient numerical computation of the electrical potential with finite element methods in 3-D and arbitrary topography. The crucial innovation is, firstly, the incorporation of unstructured tetrahedral meshes, which allow for efficient local mesh refinement and most flexible description of arbitrary model geometry. Secondly, by implementation of quadratic shape functions we achieve considerably more accurate results. Exploiting a secondary potential (SP) approach, meshes are downsized significantly in comparison with highly refined meshes for total potential calculation. However, the latter is necessary for the determination of the required primary potential in arbitrary model domains. To start with, we concentrate on the simulation of homogeneous models with different geometries at the surface and subsurface to quantify their influence. This results in a so-called geometry effect, which is not only a side effect but may be responsible for serious misinterpretations. Moreover, it represents the basis for treating heterogeneous conductivity models with the SP approach, which is especially promising for the inverse problem. We address how the resulting system of equations is solved most efficiently using modern multifrontal direct solvers in conjunction with reordering strategies or rather traditional pre-conditioned conjugate gradient methods depending on the size of the problem. Furthermore, we present a reciprocity approach to estimate modelling errors and investigate to which degree the model discretization has to be refined to yield sufficiently accurate results.
Rücker et al. (Thu,) studied this question.