This work introduces and analyzes novel stable Petrov-Galerkin EnrichedMethods (PGEM) for the Darcy problem based on the simplest but unstable continuous P1/P0 pair. Stability is recovered inside a Petrov-Galerkin framework where element-wise dependent residual functions, named multi-scale functions, enrich both velocity and pressure trial spaces. Unlike the velocity test space that is augmented with bubble-like functions, multi-scale functions correct edge residuals as well. The multi-scale functions turn out to be the well-known lowest order Raviart-Thomas basis functions for the velocity and discontinuous quadratics polynomial functions for the pressure. The enrichment strategy suggests the way to recover the local mass conservation property for nodal-based interpolation spaces. We prove that the method and its symmetric version are well-posed and achieve optimal error estimates in natural norms. Numerical validations confirm claimed theoretical results.
- Darcy model
- enriched space
- simplest element
- Petrov-Galerkin approach
Barrenechea, G. R., Franca, L. P., & Valentin, F. (2009). A symmetric nodal conservative finite element method for the Darcy equation. SIAM Journal on Numerical Analysis, 47(5), 3652-3677. https://doi.org/10.1137/080724381