Aubin–Nitsche-type explanation of L2 super-convergence

Determine whether the super-convergence observed in the L2 norm of the error for the bound-preserving and conservative enriched Galerkin method with over-penalized jumps (β ≥ 4) applied to the elliptic reaction–diffusion problem ∫Ω ε ∇u·∇v + μ u v = ∫Ω f v with Dirichlet boundary conditions can be rigorously justified by a proof strategy that mimics the Aubin–Nitsche duality argument.

Background

The paper introduces a bound-preserving and conservative enriched Galerkin (EG) method for elliptic reaction–diffusion problems that employs a nonlinear projection and over-penalization of jumps to ensure both local conservation and adherence to physical bounds. The analysis establishes existence and optimal-order convergence, and numerical experiments corroborate the theoretical results.

In the conclusion, the authors explicitly single out the question of explaining an apparent super-convergence behavior in the L2 norm of the error. A classical approach to deriving enhanced L2 estimates for linear symmetric problems is the Aubin–Nitsche duality argument. However, the present method is nonlinear due to the projection and includes over-penalization, making a direct application of standard duality techniques nontrivial. Thus, rigorously linking the observed L2 behavior to an Aubin–Nitsche-type proof remains an explicit open issue.