Stabilized Lagrange Interpolation Collocation Method: A meshfree method incorporating the advantages of finite element method

Abstract

Since most approximation functions in meshfree methods are rational functions which do not possess the Kronecker delta property, how to achieve exact integration and accurately impose the essential boundary conditions are two typical difficulties for meshfree methods. In this paper, a new stabilized Lagrange interpolation collocation method (SLICM) is proposed in which the Lagrange interpolation (LI) is employed for the approximation in a meshfree method. This method can satisfy the high order integration constraints which can conserves the high order consistency conditions in the integration form. This property leads to the exact integration in the subdomains and optimal convergence for the proposed method. Meanwhile, performing the integration in subdomains can also reduce the condition number of discrete matrix, which improves the stability of the algorithm. Since the Lagrange interpolation approximation has Kronecker delta property, the essential boundary conditions can be simply and exactly imposed like the finite element method, which further improves the accuracy of this method. Convergence studies present that the same convergence rate can be attained for utilizing the odd and even order LI shape functions, while the convergence rate is reduced if the odd order basis function is employed in the reproducing kernel (RK) approximation. Numerical examples validate the high accuracy and convergence as well as good stability of the presented method, which can outperform the direct collocation method and the stabilized collocation method based on RK approximation.