Global and local pointwise error estimates for finite element approximations to the Stokes problem on convex polyhedra

Abstract

The main goal of the paper is to show new stability and localization results for the finite element solution of the Stokes system in W 1,∞ and L ∞ norms under standard assumptions on the finite element spaces on quasi-uniform meshes in two and three dimensions. Although interior error estimates are well-developed for the elliptic problem, they appear to be new for the Stokes system on unstructured meshes. To obtain these results we extend previously known stability estimates for the Stokes system using regularized Green's functions.

Weak discrete maximum principle of finite element methods in convex polyhedra

“…By utilizing the weak maximum principle, Schatz also established the stability of the Ritz projection in L ∞ and W 1,∞ norms. Such stability results have a wide range of applications, for example to pointwise error estimates of finite element methods for parabolic problems [16,20,21], Stokes systems [3], nonlinear problems [10,11,22], obstacle problems [6], optimal control problems [1,2], to name a few. As far as we know, [25] is the only paper that establishes weak maximum principle and L ∞ stability estimate (without the logarithmic factor) for the Ritz projection on nonsmooth domains.…”

“…Let S h be a finite element space of piecewise polynomials of degree r 1 subject to a quasi-uniform tetrahedral partition of a convex polyhedron Ω ⊂ R 3 , where h denotes the mesh size of the tetrahedral partition. Let Sh be the subspace of S h consisting of functions with zero boundary values.…”