2022/08/25 by Tomoya Kemmochi, Kemmochi, Tomoya
Computer Science · Engineering · Mathematics · #47A10 #65M60 #65N30 #76D07 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2208.11892
openalex publication_date 2022/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we will show the Lp-resolvent estimate for the finite element approximation of the Stokes operator for p ∈ ( (2N)/(N+2), (2N)/(N-2) ), where N ≥ 2 is the dimension of the domain. It is expected that this estimate can be applied to error estimates for finite element approximation of the non-stationary Navier--Stokes equations, since studies in this direction are successful in numerical analysis of nonlinear parabolic equations. To derive the resolvent estimate, we introduce the solution of the Stokes resolvent problem with a discrete external force. We then obtain local energy error estimate according to a novel localization technique and establish global Lp-type error estimates. The restriction for p is caused by the treatment of lower-order terms appearing in the local energy error estimate. Our result may be a breakthrough in the Lp-theory of finite element methods for the non-stationary Navier--Stokes equations.