The Least Squares Finite Element Method for Elasticity Interface Problem on Unfitted Mesh
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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.
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