Nitsche's method under a semi-regular mesh condition

TL;DR

This paper analyzes Nitsche's method on anisotropic meshes, providing new error estimates and a novel proof technique that enhances understanding of boundary condition enforcement in finite element methods.

Contribution

It introduces a new proof for the consistency term in Nitsche's method, enabling anisotropic consistency error estimates on convex domains.

Findings

Error estimates in the energy norm for anisotropic meshes

Validation of theoretical results through numerical experiments

Enhanced understanding of boundary condition enforcement

Abstract

Nitsche's method is a numerical approach that weakly enforces boundary conditions for partial differential equations. In recent years, Nitsche's method has experienced a revival owing to its natural application in modern computational methods, such as the cut and immersed finite element methods. This study investigates Nitsche's methods based on an anisotropic weakly over-penalized symmetric interior penalty method for Poisson and Stokes equations on convex domains. As our primary contribution, we provide a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relationship between the Crouzeix and Raviart finite element space and the Raviart--Thomas finite element space. We present the error estimates in the energy norm on anisotropic meshes. We compared the calculation results for the anisotropic mesh partitions in the numerical experiments.