Abstract
This work introduces and analyzes novel stable Petrov–Galerkin enriched methods (PGEM) for the Darcy problem, based on the simplest but unstable continuous $\mathbb{P}_1/\mathbb{P}_0$ pair. Stability is recovered inside a Petrov–Galerkin framework where elementwise dependent residual functions, named multiscale functions, enrich both velocity and pressure trial spaces. Unlike the velocity test space that is augmented with bubble-like functions, multiscale functions correct edge residuals as well. The multiscale 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.
Accepted Version (
Free)
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call