Isogeometric discretizations of the Stokes problem on trimmed geometries

TL;DR

This paper investigates the stability and accuracy of isogeometric discretizations of the Stokes problem on trimmed geometries, addressing boundary condition challenges and proposing stabilized methods to ensure well-posedness and optimal error estimates.

Contribution

It extends stabilization techniques to incompressible flows in trimmed geometries, ensuring stability and optimal convergence of isogeometric discretizations.

Findings

Nitsche method can be unstable in degenerate configurations

Stabilization restores well-posedness and stability

Numerical experiments confirm convergence rates

Abstract

The isogeometric approximation of the Stokes problem in a trimmed domain is studied. This setting is characterized by an underlying mesh unfitted with the boundary of the physical domain making the imposition of the essential boundary conditions a challenging problem. A very popular strategy is to rely on the so-called Nitsche method \cite{MR3264337}. We show that the Nitsche method lacks stability in some degenerate trimmed domain configurations, potentially polluting the computed solutions. After extending the stabilization procedure of \cite{MR4155233} to incompressible flow problems, we show that we recover the well-posedness of the formulation and, consequently, optimal a priori error estimates. Numerical experiments illustrating stability and converge rates are included.