A Unified Transmissibility-Based Interior Penalty DG Method for Heterogeneous and Anisotropic Diffusion

TL;DR

This paper introduces a unified interior penalty discontinuous Galerkin method for complex heterogeneous and anisotropic diffusion problems, ensuring stability and accuracy regardless of material contrasts.

Contribution

The authors develop a novel unified DG scheme that combines various interior penalty approaches with transmissibility-based weights, enhancing robustness and theoretical guarantees.

Findings

The method is stable for high contrast and anisotropy.

The scheme achieves quasi-optimal error estimates.

Numerical results confirm theoretical stability and accuracy.

Abstract

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.