$L^p$-resolvent estimate for finite element approximation of the Stokes operator

TL;DR

This paper establishes $L^p$-resolvent estimates for finite element approximations of the Stokes operator, which could significantly impact error analysis for non-stationary Navier--Stokes equations.

Contribution

It introduces a novel localization technique to derive $L^p$-resolvent estimates for finite element methods applied to the Stokes operator, extending the theoretical framework.

Findings

Established $L^p$-resolvent estimates for finite element Stokes approximation.

Developed a new localization technique for error analysis.

Potential application to error estimates in Navier--Stokes simulations.

Abstract

In this paper, we will show the $L^p$-resolvent estimate for the finite element approximation of the Stokes operator for $p \in \left( \frac{2N}{N+2}, \frac{2N}{N-2} \right)$, where $N \ge 2$ 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 $L^p$-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 $L^p$-theory of finite element methods for the non-stationary Navier--Stokes equations.