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

Abstract

We analyse a non-conforming finite-element method to approximate advection-diffusion-reaction equations. The method is stabilized by penalizing the jumps of the solution and those of its advective derivative across mesh interfaces. The a priori error analysis leads to (quasi-)optimal estimates in the mesh size (sub-optimal by order 1 2 in the L 2 -norm and optimal in the broken graph norm for quasi-uniform meshes) keeping the Péclet number fixed. Then, we investigate a residual a posteriori error estimator for the method. The estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Péclet number. Finally, to illustrate the theory we present numerical results including adaptively generated meshes.

Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods

“…His estimates are both reliable and locally efficient and, moreover, the efficiency constant becomes optimal as the local Péclet number gets sufficiently small. Similar results have been obtained in the framework of nonconforming finite element methods by Ainsworth [7] for the inhomogeneous pure diffusion case and by El Alaoui et al in [26] for the convection-diffusion case. Recently, Verfürth [47] improved his results while giving estimates which are fully robust with respect to convection dominance in a norm incorporating a dual norm of the convective derivative.…”

“…This problem is a modification of a problem considered in [26]. We put Ω = (0, 1)×(0, 1), w = (0, 1), and r = 1 in (1.1a) and consider three cases with S = ε Id and ε equal to, respectively, 1, 10 −2 , and 10 −4 .…”

Robust a posteriori error estimates for conforming and nonconforming finite element methods for convection–diffusion problems

“…The nonconforming finite element method with face penalty was introduced in [15] and the corresponding error analysis was presented therein. For convenience, we set…”

“…In [14], El Alaoui and Ern designed and analyzed a nonconforming finite element method with subgrid viscosity to solve the convection-diffusion problem, which provided the first extension of the subgrid viscosity technique introduced by Guermond [18,19] to nonconforming settings. The nonconforming finite element methods with face and interior penalty were discussed in [4,15].…”

“…El Alaoui et al [15] derived a semi-robust error estimator for the nonconforming finite element discretization with face penalty. In our latest work [38], semi-robust a posteriori error estimation for the nonconforming streamline-diffusion methods [22] was presented, and the error analysis was also shown to be applied to other nonconforming finite element methods with face penalty [15] and subgrid viscosity [14]. This can be viewed as a first step towards establishing robust error estimates in the nonconforming setting.…”