CutIGA with Least Squares Stabilized Nitsche Boundary Conditions: The Biharmonic Problem

We propose a symmetric Nitsche method for the biharmonic Dirichlet problem on a smooth domain whose boundary may cut arbitrarily through the computational mesh. Bulk least-squares stabilization in a boundary strip and tangential boundary penalties control the consistency terms through an explicit lifting of the two Dirichlet traces. This gives coercivity on $H^4(\mathrm{\Omega })$ without inverse inequalities and permits a fixed penalty independent of the cut configuration for sufficiently thin strips. Under suitable regularity assumptions, splines of degree $p\ge 4$ give energy and $L^2$ error bounds of orders $h^{p-1}$ and $h^{p+1}$. We also analyze a face-stabilized extension to $C^1$ splines of degree $p=2$ and $C^2$ splines of degree $p=3$. Exact-ellipse experiments examine convergence, cut stability, and parameter sensitivity.