Modeling techniques commonly exhibit errors of 3 to 10 percent or more in the calculation of apparent resistivities over earth models for which analytic solutions are easily available. A singularity occurs in the solution of any elliptic partial differential equation for which the forcing function is not smooth. The inability to adequately represent in discrete space a discontinuous function (in this case, the delta function describing the introduction of current at a point) commonly results in numerical error near the source of a modeled singularity.Inspection of an integrated finite-difference method for modeling the de resistivity geophysical technique indicates much of the error encountered is of singular origin. A procedure is herein detailed by which the singularity is mathematically removed from the modeling process and reintroduced as a last step, thus preventing it from contributing to the numerical error. Using this procedure, the average error in apparent resistivity values for a model of a polar-dipole traverse over a nonconducting sphere is reduced by 40 percent. For a dipole-dipole traverse of a two-layer model the error decreases by 75 percent, and in the case of a Wenner profile of a model of a vertically faulted earth, the average error is diminished by 90 percent.
We present a scheme for solving two-dimensional, nonlinear reaction-diffusion equations,using a mixed finite-element method. To linearize the mixed-method equations, we use a two grid scheme that relegates all the Newton-like iterations to a grid H much coarser than the original one h , with no loss in order of accuracy so long as the mesh sizes obeyThe use of a multigrid-based solver for the indefinite linear systems that arise at each coarse-grid iteration, as well as for the similar system that arises on the fine grid, allows for even greater efficiency.
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