Fictitious domain method with boundary value correction using penalty-free Nitsche method

TL;DR

This paper introduces a penalty-free Nitsche-based fictitious domain method with boundary value correction and ghost penalty stabilization, achieving optimal error estimates and supporting high-order geometry approximation.

Contribution

It develops a novel fictitious domain approach combining boundary value correction and ghost penalty stabilization without penalty terms, enabling high-order geometry approximation and proven error bounds.

Findings

Optimal error estimates in H^1-norm.

Suboptimal L^2-norm estimates due to lack of adjoint consistency.

Numerical results confirm theoretical predictions.

Abstract

In this paper, we consider a fictitious domain approach based on a Nitsche type method without penalty. To allow for high order approximation using piecewise affine approximation of the geometry we use a boundary value correction technique based on Taylor expansion from the approximate to the physical boundary. To ensure stability of the method a ghost penalty stabilization is considered in the boundary zone. We prove optimal error estimates in the $H^1$-norm and estimates suboptimal by $\mathcal{O}(h^{\frac12})$ in the $L^2$-norm. The suboptimality is due to the lack of adjoint consistency of our formulation. Numerical results are provided to corroborate the theoretical study.