A Unified Transmissibility-Based Interior Penalty DG Method for Heterogeneous and Anisotropic Diffusion
This work provides a theoretical unification and robustness guarantee for DG methods in diffusion problems with high contrast, benefiting computational scientists solving subsurface flow or similar problems.
The authors derive a unified interior penalty DG method for heterogeneous and anisotropic diffusion that is robust to high contrast and anisotropy, with stability independent of diffusion variations. Numerical experiments confirm quasi-optimal error estimates.
We derive a primal discontinuous Galerkin (DG) formulation for heterogeneous and anisotropic diffusion, obtained by exact algebraic elimination of the skeletal unknown in a compact hybridized interior penalty (H-IP) method. The resulting Unified Interior Penalty DG (UIP-DG) scheme involves transmissibility-based weights inherited from the hybrid formulation, together with two stabilization terms acting respectively on the primal jump and on the jump of the normal diffusive flux. These penalties scale, respectively, with the harmonic mean and with the inverse arithmetic mean of the face-wise transmissibilities. This construction provides a unified perspective on several interior penalty approaches previously introduced independently, while yielding a robust method with stability properties independent of the diffusion contrast and anisotropy. We prove consistency, coercivity, and boundedness of the formulation, and derive quasi-optimal energy-norm a priori error estimates for all variants. Numerical experiments confirm the theoretical claims.