A bound-preserving and conservative enriched Galerkin method for elliptic problems

We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial over-penalization of the discrete solution’s jumps to obtain optimal convergence. To avoid the ill-conditioning issues that arise in over-penalized schemes, we introduce an involved splitting approach that separates the system of equations for the discontinuous solution part from the system of equations for the continuous solution part, yielding well-behaved subproblems. We prove the existence of discrete solutions and optimal error estimates, which are validated numerically.

💡 Research Summary

This paper introduces a locally conservative enriched Galerkin (EG) discretization for linear elliptic reaction‑diffusion problems that simultaneously preserves the physical bounds of the exact solution. The authors observe that standard EG methods, while offering optimal approximation properties and local mass conservation, do not guarantee that the discrete solution stays within the invariant interval (