Local pointwise a posteriori gradient error bounds for the Stokes equations

Abstract

Abstract. We consider the standard Taylor-Hood finite element method for the stationary Stokes system on polyhedral domains. We prove local a posteriori error estimates for the maximum error in the gradient of the velocity field. Because the gradient of the velocity field blows up near re-entrant corners and edges, such local error control is necessary when pointwise control of the gradient error is desirable. Computational examples confirm the utility of our estimates in adaptive codes.

An Adaptive FEM for the Pointwise Tracking Optimal Control Problem of the Stokes Equations

“…The following result state a global higher integrability result for the solution (y, p) and, as consequence, a Hölder regularity estimate for the velocity field y; see [13, Theorem 2.9], [34, Theorem 1.1] and [18,Lemma 12].…”

Maximum–norm a posteriori error estimates for an optimal control problem

“…Proof Let v ∈ H 1 0 (Ω ) be such that v |T ∈ C 2 (T ) for all T ∈ T . Using (11) and integrating by parts yields…”

“…The results of Nochetto [33] were later extended to three-dimensional domains in [7] and subsequently improved in [9,10,34]. The theory has also been extended to problems involving unbounded forcing terms, the Stokes equations, and obstacle, monotone semilinear, and geometric problems [3,6,10,11,34,35].…”

A diffuse interface model for two-phase ferrofluid flows

“…[39,44,27,46,38]), and is limited to a handful of finite element pairs {U h , P h }, such as the second and third order Taylor-Hood element, and the lowest order Bernardi-Raugel element. All the max-norm estimates reported in [39,44,27,46,38] use finite element pairs {U h , P h } with continuous velocities combined with continuous pressures, or discontinuous pressures of order zero (piecewise constants), which do not satisfy the assumption (A1) of our list above. We are not aware of max-norm error estimates for stable finite element pairs {U h , P h } using continuous velocities U h and higher-order (at least first-order) discontinuous pressures P h , thus, satisfying (A1).…”