Adaptive Consistency in Mimetic Finite Difference Schemes: A Residual-Based Criterion for K-orthogonal Cell Classification
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Mimetic Finite Difference (MFD) schemes ensure local conservation on general polytopal meshes but become computationally expensive when applied uniformly. In practice, many well-shaped cells satisfy the Two-Point Flux Approximation (TPFA) K-orthogonality consistency requirement, rendering full MFD stencils unnecessary on large portions of the grid. To improve efficiency, we present a residual-driven local consistency indicator for adaptive TPFA-MFD schemes. The proposed indicator evaluates operator consistency by projecting an analytical linear pressure field onto the discrete degrees of freedom and measuring the resulting cell-local residual of the MFD weak formulation. The approach is purely local, requiring no global assembly or auxiliary solvers; it relies solely on the geometry and absolute permeability associated with each cell. Validated on 2D and 3D polytopal meshes, the criterion reliably distinguishes TPFA-compatible cells from those affected by distortion or anisotropy. By enabling a priori cell classification, the methodology reduces computational overhead and improves matrix sparsity while maintaining accuracy on complex geometries, providing a scalable tool for large-scale porous media simulations.
