A Nonlocal Biharmonic Model with $\mathrm{\Gamma }$-Convergence to Local Model and an Efficient Numerical Method

Nonlocal models and their associated theories have been extensively investigated in recent years. Among these, nonlocal versions of the classical Laplace operator have attracted the most attention, while higher-order nonlocal operators have been studied far less. In this work, we focus on the nonlocal counterpart of the classical biharmonic operator together with the clamped boundary condition ($u$ and $\frac{\mathrm{\partial }u}{\mathrm{\partial }n}$ are given on the boundary). We develop the variational formulation of a nonlocal biharmonic model, establish the existence and uniqueness of its solution, and analyze its convergence as the nonlocal horizon approaches zero. In addition, we propose an efficient finite element method to solve the nonlocal model and the numerical results verify the analytical properties of the nonlocal model and its solution.