Local error estimates for finite element discretization of the Stokes equations

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

“…Roughly speaking, the local L 2 analysis shows that the error for both the pressure and the gradient of the velocity measured by the L 2 (D 0 ) − norm for a subdomain D 0 ⊂ Ω is bounded by the best approximation error in the L 2 (D 1 ) − norm for a slightly larger subdomain D 1 plus the error in a weaker norm. These estimates are very similar to the local error estimates obtained by Arnold and Liu [2] for conforming mixed methods applied to (1.1). However, the results in [2] are for interior subdomains D 0 , whereas in this paper we allow D 0 to touch ∂Ω.…”

“…These estimates are very similar to the local error estimates obtained by Arnold and Liu [2] for conforming mixed methods applied to (1.1). However, the results in [2] are for interior subdomains D 0 , whereas in this paper we allow D 0 to touch ∂Ω. Many of the techniques to prove local error estimates presented in this paper and in [2] are borrowed from the techniques developed by Nitsche and Schatz [20] for proving local estimates of conforming finite element methods for the Laplace equation.…”

“…In this paper we also use the techniques found in [21] and our results are very similar to the results contained in [3]. However, in order to prove pointwise estimates Chen assumed local error estimates for subdomains that touch ∂Ω which are not contained in [2]. As mentioned above, in this paper we prove local estimates for subdomains that touch ∂Ω for the LDG method.…”

“…However, the results in [2] are for interior subdomains D 0 , whereas in this paper we allow D 0 to touch ∂Ω. Many of the techniques to prove local error estimates presented in this paper and in [2] are borrowed from the techniques developed by Nitsche and Schatz [20] for proving local estimates of conforming finite element methods for the Laplace equation. However, the pressure term and the incompressibility equation add extra difficulties when analyzing the Stokes problem.…”

“…These pointwise estimates are optimal and describe how the error at a point x depends on the behavior of the exact solution in regions away from x. Recently, Chen [3] used the local estimates derived in [2] to prove pointwise estimates of conforming mixed methods for (1.1) on a domain Ω with a smooth boundary. Chen makes use of techniques originally developed by Schatz [21] to prove pointwise estimates for the Laplace equation.…”