[Paper Review] A Cut Finite Element Method with Boundary Value Correction

Erik Burman, Peter Hansbo+1 more|arXiv (Cornell University)|Jul 11, 2015

Advanced Numerical Methods in Computational Mathematics8 references4 citations

TL;DR

This paper presents a cut finite element method with boundary value correction that enables optimal convergence using only piecewise linear boundary approximations, overcoming geometric complexity in fictitious domain methods. By incorporating a Taylor expansion in the normal direction to correct boundary conditions, the method achieves optimal error estimates in energy and $L^2$ norms without requiring high-accuracy boundary representation.

ABSTRACT

In this contribution we develop a cut finite element method with boundary value correction of the type originally proposed by Bramble, Dupont, and Thomee. The cut finite element method is a fictitious domain method with Nitsche type enforcement of Dirichlet conditions together with stabilization of the elements at the boundary which is stable and enjoy optimal order approximation properties. A computational difficulty is, however, the geometric computations related to quadrature on the cut elements which must be accurate enough to achieve higher order approximation. With boundary value correction we may use only a piecewise linear approximation of the boundary, which is very convenient in a cut finite element method, and still obtain optimal order convergence. The boundary value correction is a modified Nitsche formulation involving a Taylor expansion in the normal direction compensating for the approximation of the boundary. Key to the analysis is a consistent stabilization term which enables us to prove stability of the method and a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.

Motivation & Objective

• To address the computational challenge of accurate quadrature on cut elements in cut finite element methods (CutFEM).
• To enable optimal convergence rates when using only piecewise linear approximations of the boundary, simplifying geometric computations.
• To develop a stabilized Nitsche formulation with boundary value correction that maintains stability and optimal convergence despite boundary approximation errors.
• To prove a priori error estimates with explicit dependence on meshsize and boundary approximation error.
• To demonstrate that optimal convergence is achievable even with low-order boundary representation, avoiding the need for $O(h^{p+1})$ boundary accuracy required in standard CutFEM.

Proposed method

• Introduces a boundary value correction via a Taylor expansion in the normal direction to approximate the solution on the exact boundary using values and derivatives at the approximate boundary.
• Uses a Nitsche-type weak enforcement of Dirichlet conditions on the approximate boundary, modified by the Taylor expansion to correct for geometric errors.
• Applies a consistent stabilization term that controls function variation near the boundary, ensuring coercivity and optimal condition number $O(h^{-2})$.
• Employs the closest point mapping to define the exact boundary and enables consistent integration over the approximate domain.
• Uses a piecewise linear distance function to define the discrete domain, simplifying geometric computation.
• Derives and analyzes a symmetric variational formulation that combines Nitsche's method with boundary correction and stabilization.

Experimental results

Research questions

• RQ1 Can optimal convergence be achieved in CutFEM using only a piecewise linear approximation of the boundary, avoiding high-order geometric reconstruction?
• RQ2 How does the inclusion of a Taylor expansion in the normal direction improve the accuracy of Dirichlet boundary conditions in cut finite element methods?
• RQ3 What stabilization mechanism ensures stability and optimal conditioning when the boundary cuts through elements arbitrarily?
• RQ4 Does the proposed method maintain optimal error estimates in both energy and $L^2$ norms under boundary approximation errors?
• RQ5 Can the method achieve optimal convergence for higher-order polynomial approximations (e.g., $P^2$, $P^3$) with minimal Taylor expansion terms?

Key findings

• The method achieves optimal order convergence in both energy and $L^2$ norms, even when the boundary is approximated only by a piecewise linear representation.
• For $P^2$ and $P^3$ elements, only the first two terms of the Taylor expansion are needed to achieve optimal convergence rates.
• The stabilization term ensures that the condition number of the linear system remains $O(h^{-2})$, independent of the boundary's position relative to the background mesh.
• Numerical results confirm optimal convergence rates of 2 for $P^2$ and 4 for $P^3$ elements, matching theoretical predictions.
• The method maintains stability and accuracy without requiring extension of the discrete solution beyond the active domain, even when elements are partially cut.
• The error estimates are explicit in terms of the meshsize $h$ and the distance between the exact and approximate boundaries, enabling precise convergence analysis.