Nonconforming finite element methods with subgrid viscosity applied to advection‐diffusion‐reaction equations

Abstract

A nonconforming (Crouzeix-Raviart) finite element method with subgrid viscosity is analyzed to approximate advection-diffusion-reaction equations. The error estimates are quasi-optimal in the sense that keeping the Péclet number fixed, the estimates are suboptimal of order 1 2 in the mesh size for the L 2 -norm and optimal for the advective derivative on quasi-uniform meshes. The method is also reformulated as a finite volume box scheme providing a reconstruction formula for the diffusive flux with local conservation properties. Numerical results are presented to illustrate the error analysis.

A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection–diffusion equations

“…Second, Crouzeix-Raviart finite elements have close links with finitevolume box schemes; see, e.g. Courbet & Croisille (1998) and Croisille (2000) for Darcy's equations and El Alaoui & Ern (2006) for advection-diffusion equations. This property is useful to reconstruct locally the diffusive flux in problems where conservativity properties are important, e.g.…”

Semidiscrete formulations for transient transport at small time steps

“…Note that in (20), the Galerkin predictor diffusion contribution on the right-hand side emanates naturally in integrated-by-parts form. This has an important impact on accuracy and the same philosophy should be applied to the other predictor viscous terms on the stabilization integrals.…”

“…In order to stabilize the Galerkin discretization (20), the combined adjoint stabilization method [21] is implemented. This method adds two stabilization integrals to the Galerkin method (a leastsquares plus a gradient least-squares term), and can be interpreted as an approximation to the exact variational multiscale method, where the cross moments of the element Green's function have been neglected.…”

“…Any scheme that stabilizes the steady advection-diffusionreaction equation, e.g. [19][20][21][22] removes this pathology. Similar ideas are explored in the context of enriched finite elements based on a multiscale approach [23].…”

Nonconforming local projection stabilization for generalized Oseen equations

“…However, the above methods adopt conforming finite element spaces in general. As far as we know, there is only one paper adopting nonconforming finite element methods with subgrid viscosity applied to the advection-diffusion-reaction equations [15] . Kaya and Rivière proposed a discontinuous subgrid eddy viscosity method for the time-dependent Navier-Stokes equations [12] .…”