Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra

Abstract

The aim of the paper is to show the stability of the finite element solution for the Stokes system in W 1 ∞ norm on general convex polyhedral domain. In contrast to previously known results, W 2 r regularity for r > 3, which does not hold for general convex polyhedral domains, is not required. The argument uses recently available sharp Hölder pointwise estimates of the corresponding Green's matrix together with novel local energy error estimates, which do not involve an error of the pressure in a weaker norm.

Max-Norm Stability of Low Order Taylor–Hood Elements in Three Dimensions

“…Second, we cannot use directly the local energy estimates that Chen assumed because this will require us to estimate the pressure error in a negative order norm which we do not know how to estimate with the given regularity of the problem. Instead we prove a local energy estimate that does not contain the error of the pressure which is very similar to the estimates obtained in [11] (see also [12]). Of course, the estimates derived in [11] assumed the existence of a quasi-local Fortin projection.…”

“…Instead we prove a local energy estimate that does not contain the error of the pressure which is very similar to the estimates obtained in [11] (see also [12]). Of course, the estimates derived in [11] assumed the existence of a quasi-local Fortin projection.…”

“…In previous papers [8,11], W 1,∞ × L ∞ stability was proven for many inf-sup stable pair of spaces, but one major exception was the lowest order Taylor-Hood pair in three dimensions (k = 2). The reason for this is that in both papers it was assumed that there exists a Fortin projection h (i.e.…”

Local pointwise a posteriori gradient error bounds for the Stokes equations

“…The disadvantage of this approach is that it requires an unnatural restriction on the maximum interior dihedral angle when n = 3. The a priori estimates of [8], [9] suffer from the same restriction, which was subsequently overcome in [10] by the use of sharp Green's function estimates. We similarly avoid this restriction by employing sharp Green's function estimates.…”

“…Local a priori error estimates for the Stokes equation are developed in [12] and used to justify a local parallel finite element algorithm. Global W 1 ∞ a priori error estimates for the Stokes system on convex polygonal and polyhedral domains can be found in the recent papers [8], [9], and [10]; we also refer to these works for a more comprehensive overview of previous literature on maximum norm a priori analysis for the Stokes problem.…”