Vincent Fontaine | Searcharxiv

arXiv2026

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

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.

math.NA↗

arXiv2021

Improved error estimates of hybridizable interior penalty methods using a variable penalty for highly anisotropic diffusion problems

In this paper, we derive improved a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a variable penalty for second-order elliptic problems. The strategy is to use a penalization function of the form $\mathcal{O}(1/h^{1+δ})$, where $h$ denotes the mesh size and $δ$ is a user-dependent parameter. We then quantify its direct impact on the convergence analysis, namely, the (strong) consistency, discrete coercivity, and boundedness (with $h^δ$-dependency), and we derive updated error estimates for both discrete energy- and $L^{2}$-norms. The originality of the error analysis relies specifically on the use of conforming interpolants of the exact solution. All theoretical results are supported by numerical evidence.

math.NA↗