The Least Squares Finite Element Method for Elasticity Interface Problem on Unfitted Mesh
share
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 C^2 or polygonal, and the exact solution (σ,u) belongs to H^s(div; Ω_0 ∪Ω_1) ×H^1+s(Ω_0 ∪Ω_1)withs > 1/2. Two types of least squares functionals are defined to seek the numerical solution. The first is defined by simply applying theL^2norm least squares principle, and requires the conditions ≥1. The second is defined with a discrete minus norm, which is related to the inner product inH^-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 anys > 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 underL^2norm 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.
READ FULL TEXT
