Adaptive finite-element modeling using unstructured grids: The 2D magnetotelluric example

Existing numerical modeling techniques commonly used for electromagnetic (EM) exploration are bound by the limitations of approximating complex structures using a rectangular grid. A more flexible tool is the adaptive finite-element (FE) method using unstructured grids. Composed of irregular triangles, an unstructured grid can readily conform to complicated structural boundaries. To ensure numerical accuracy, adaptive refinement using an a posteriori error estimator is performed iteratively to refine the grid where solution accuracy is insufficient. Two recently developed asymptotically exact a posteriori error estimators are based on a superconvergent gradient recovery operator. The first relies solely on the normed difference between the recovered gradients and the piecewise constant FE gradients and is effective for lowering the global error in the FE solution. For many problems, an accurate solution is required only in a few discrete regions and a more efficient error estimator is possible by considering the local influence of errors from coarse elements elsewhere in the grid. The second error estimator accomplishes this by using weights determined from the solution to an appropriate dual problem to modify the first error estimator. Application of these methods for 2D magnetotelluric (MT) modeling reveals, as expected, that the dual weighted error estimator is far more efficient in achieving accurate MT responses. Refining about 15% of elements per iteration gives the fastest convergence rate. For a given refined grid, the solution error at higher frequencies varies in proportion to the skin depth, requiring refinement about every two decades of frequency. The transverse electric (TE) and transverse magnetic (TM) modes exhibit different field behavior, and refinement should consider the effects of both. An example resistivity model of seafloor bathymetry underlain by complex salt intrusions and dipping and faulted sedimentary layers illustrates the benefits of this new technique.