New error estimates of nonconforming mixed finite element methods for the Stokes problem

Abstract

In this paper, we revisit the classical error estimates of nonconforming Crouzeix–Raviart type finite elements for the Stokes equations. By introducing some quasi‐interpolation operators and using the special properties of these nonconforming elements, it is proved that their consistency errors can be bounded by their approximation errors together with a high‐order term, especially which can be of arbitrary order provided that f in the right‐hand side is piecewise smooth enough. Furthermore, we show an interesting result that both in the energy norm and L2 norm the consistency errors are dominated by the approximation errors of their finite element spaces. As byproducts, we derive the error estimates in both energy and L2 norms under the regularity assumption (u,p) ∈ H1 + s(Ω) × Hs(Ω) with any s ∈ (0,1], which fills the gap in the a priori error estimate of these nonconforming elements with low regularity (bold-italicuMathClass-punc,p)MathClass-rel∈boldH1MathClass-bin+s(Ω)MathClass-bin×Hs(Ω)MathClass-punc,0MathClass-rel<sMathClass-rel≤12. Furthermore, a robust convergence is proved with minimal regularity assumption s = 0. These results seem to be missing in the literature. Numerical tests are provided, confirming the analysis, especially the new results on the L2 convergence. Copyright © 2013 John Wiley & Sons, Ltd.

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

“…In this subsection, we study the convergence analysis of the scheme when the exact solution does not possess the full regularity, i.e., when u ∈ H s (Ω) for some s < 3 only. Similar to the discussion in [33], the a priori error estimate for solutions with the low regularity can be useful for the definition of nonlinear approximation classes and for the analysis of the quasi-optimality of adaptive finite element methods for the biharmonic equation. Again, we begin with an estimate for an auxiliary Stokes problem.…”

Analysis of pressure-robust embedded-hybridized discontinuous Galerkin methods for the Stokes problem under minimal regularity

“…To put the error analysis of this paper into a broader context, we review briefly the literature relevant to our analysis. Numerous classes of non-pressure-robust nonconforming methods for the Stokes problem were analyzed under minimal regularity assumptions in [2,27]. Key to the analysis of [2,27] is a so-called enrichment operator that maps nonconforming discrete functions to H 1 -conforming functions.…”

“…Numerous classes of non-pressure-robust nonconforming methods for the Stokes problem were analyzed under minimal regularity assumptions in [2,27]. Key to the analysis of [2,27] is a so-called enrichment operator that maps nonconforming discrete functions to H 1 -conforming functions. More recently, by using enrichment operators that map discretely divergence-free functions to exactly divergence-free ones, this minimal regularity analysis has been extended to pressure-robust schemes.…”

A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh

“…Unlike the traditional finite elements [5,8,9,16,17,18,19,20,21], the inf-sup condition for the weak Galerkin finite element is easily satisfied due to the large velocity space with independent element boundary degrees of freedom. Lemma 3.2.…”

“…Nevertheless we can see such a computed flow is not mass conservative. We need to use H(div) finite elements [15] or divergence-free H 1 finite elements [5,8,9,16,17,18,19,20,21] to get a mass conservative solution. We use tetrahedral meshes shown in Figure 13.…”