The mimetic multiscale method for Maxwell’s equations

Abstract

We have developed a mimetic multiscale method to simulate quasistatic Maxwell’s equations in the frequency domain. This is especially useful for extensive geophysical models that include small-scale features. Applying the concept of multiscale methods, we avoid setting up a large and costly system of equations on the fine mesh where the material parameters are discretized on. Instead, we build and solve a system on a much coarser mesh. For doing that, it is inevitable to interpolate between fine and coarse meshes. The construction of this coarse-to-fine interpolation is done by solving local, frequency-independent optimization problems for the electric field and the magnetic flux on each coarse cell incorporating the fine-mesh features. Hence, the interpolation operators transfer the fine-mesh material properties onto the coarse simulation mesh. To increase the accuracy of the interpolation, we apply oversampling; i.e., the coarse-cell optimization problems are solved on extended local domains. Previous work on multiscale methods for Maxwell’s equations is not capable of keeping the mimetic properties of the discretization. With our method being mimetic, the properties of the continuous differential operators are preserved in their discrete counterparts and thus, the resulting simulations do not contain spurious modes. We determine the effectiveness of our multiscale construction with coarse-mesh simulations for two examples, a vertical borehole and a mine model.