A New Least Squares Stabilized Nitsche Method for Cut Isogeometric Analysis

TL;DR

This paper introduces a new stabilized Nitsche method for cut isogeometric analysis that enforces boundary conditions effectively, providing optimal error estimates and robust performance even in challenging cut scenarios.

Contribution

The paper presents a novel least squares stabilized symmetric Nitsche method for CutIGA, combining elementwise stabilization with basis function removal for well-posedness.

Findings

Method achieves coercivity and optimal error estimates.

Performs well in extreme cut situations.

Stabilization is consistent and only elementwise.

Abstract

We derive a new stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions for elliptic problems of second order in cut isogeometric analysis (CutIGA). We consider $C^1$ splines and stabilize the standard Nitsche method by adding certain elementwise least squares terms in the vicinity of the Dirichlet boundary and an additional term on the boundary which involves the tangential gradient. We show coercivity with respect to the energy norm for functions in $H^2(\Omega)$ and optimal order a priori error estimates in the energy and $L^2$ norms. To obtain a well posed linear system of equations we combine our formulation with basis function removal which essentially eliminates basis functions with sufficiently small intersection with $\Omega$. The upshot of the formulation is that only elementwise stabilization is added in contrast to standard procedures based on ghost penalty and related techniques and that the stabilization is consistent. In our numerical experiments we see that the method works remarkably well in even extreme cut situations using a Nitsche parameter of moderate size.