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A Finite-Element Framework for a Mimetic Finite-Difference Discretization of Maxwell's Equations
SIAM Journal on Scientific Computing  (IF2.373),  Pub Date : 2021-07-28, DOI: 10.1137/20m1382568
James H. Adler, Casey Cavanaugh, Xiaozhe Hu, Ludmil T. Zikatanov

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2638-A2659, January 2021.
Maxwell's equations are a system of partial differential equations that govern the laws of electromagnetic induction. We study a mimetic finite-difference (MFD) discretization of the equations which preserves important underlying physical properties. We show that, after mass-lumping and appropriate scaling, the MFD discretization is equivalent to a structure-preserving finite-element (FE) scheme. This allows for a transparent analysis of the MFD method using the FE framework and provides an avenue for the construction of efficient and robust linear solvers for the discretized system. In particular, block preconditioners designed for FE formulations can be applied to the MFD system in a straightforward fashion. We present numerical tests which verify the accuracy of the MFD scheme and confirm the robustness of the preconditioners.