  In order to reduce the computational cost of the simulation of
electromagnetic responses in geophysical settings that involve highly
heterogeneous media, we develop a multiscale finite volume method with
oversampling for the quasi-static Maxwell's equations in the frequency domain.
We assume a coarse mesh nested within a fine mesh that accurately discretizes
the problem. For each coarse cell, we independently solve a local version of
the original Maxwell's system subject to linear boundary conditions on an
extended domain, which includes the coarse cell and a neighborhood of fine
cells around it. The local Maxwell's system is solved using the fine mesh
contained in the extended domain and the mimetic finite volume method. Next,
these local solutions (basis functions) together with a weak-continuity
condition are used to construct a coarse-mesh version of the global problem.
The basis functions can be used to obtain the fine-mesh details from the
solution of the coarse-mesh problem. Our approach leads to a significant
reduction in the size of the final system of equations and the computational
time, while accurately approximating the behavior of the fine-mesh solutions.
We demonstrate the performance of our method using a synthetic 3D example of a
mineral deposit.
