The least squares finite element method for elasticity interface problem on unfitted mesh

College of Mathematics, Sichuan University, Chengdu, P.R. China

^* Corresponding author: yangfanyi@scu.edu.cn

Received: 17 July 2023
Accepted: 1 March 2024

Abstract

In this paper, we propose and analyze the least squares finite element methods for the linear elasticity interface problem in the stress-displacement system on unfitted meshes. We consider the cases that the interface is C² or polygonal, and the exact solution (σ, u) belongs to H^s(div; Ω₀ ∪ Ω₁) × H^1+s(Ω₀ ∪ Ω₁) with s > 1/2. Two types of least squares functionals are defined to seek the numerical solutions. The first is defined by simply applying the L² norm least squares principle, and requires the condition s ≥ 1. The second is defined with a discrete minus norm, which is related to the inner product in H^−1/2(Γ). The use of this discrete minus norm results in a method of optimal convergence rates and allows the exact solution has the regularity of any s > 1/2. The stability near the interface for both methods is guaranteed by the ghost penalty bilinear forms and we can derive the robust condition number estimates. The convergence rates under L² norm and the energy norm are derived for both methods. We illustrate the accuracy and the robustness of the proposed methods by a series of numerical experiments for test problems in two and three dimensions.