A Finite-Element Framework for a Mimetic Finite-Difference Discretization of Maxwell's Equations

Adler, James H.; Cavanaugh, Casey; Hu, Xiaozhe; Zikatanov, Ludmil T.

doi:10.1137/20M1382568

Strong coupling between geomechanical deformation and multiphase fluid flow appears in a variety of geoscience applications. A common discretization strategy for these problems is a continuous Galerkin finite element scheme for the momentum balance equation and a finite volume scheme for the mass balance equations. When applied within a fully implicit solution strategy, however, this discretization is not intrinsically stable. In the limit of small time steps or low permeabilities, spurious oscillations in the piecewise-constant pressure field, i.e., checkerboarding, may be observed. Further, eigenvalues associated with the spurious modes will control the conditioning of the matrices and can dramatically degrade the convergence rate of iterative linear solvers. Here, we propose a stabilization technique in which the mass balance equations are supplemented with stabilizing flux terms on a macroelement basis. The additional stabilization terms are dependent on a stabilization parameter. We identify an optimal value for this parameter using an analysis of the eigenvalue distribution of the macroelement Schur complement matrix. The resulting method is simple to implement and preserves the underlying sparsity pattern of the original discretization. Another appealing feature of the method is that mass is exactly conserved on macroelements, despite the addition of artificial fluxes. The efficacy of the proposed technique is demonstrated with several numerical examples.

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