DIRECT COLLOCATION METHOD AND STABILIZED COLLOCATION METHOD BASED ON LAGRANGE INTERPOLATION FUNCTION

Since most of the approximation functions in the meshfree method are rational and do not have the Kronecker delta property, it is difficult to accurately impose the essential boundary conditions. Large errors on the boundary can easily lead to low accuracy of the solution in the whole solution domain and may even introduce the numerical instability in solution process. In this paper, the Lagrange interpolation function is introduced as the shape function in the meshfree direct collocation method and the stabilized collocation method, and the Lagrange interpolation collocation method (LICM) and the stabilized Lagrange interpolation collocation method (SLICM) are constructed. Since Lagrange interpolation has the Kronecker delta property, the essential boundary conditions can be imposed as simply and precisely as the finite element method, which promotes the numerical solution accuracy of the two methods. The stabilized collocation method is based on the subdomain integration, which can satisfy the high order integration constraints. That is, it can ensure that the shape function also meets the high-order consistency conditions in the integral form and achieve accurate integration. At the same time, the subdomain integration can also reduce the condition number of the discrete matrix, which improves the stability of the algorithm. By combining the Lagrange interpolation function and the stabilized collocation method, the accuracy and stability of the stabilized Lagrange interpolation collocation method is further improved. Numerical examples validate the accuracy, convergence and stability of the proposed Lagrange interpolation collocation method (LICM) and the stabilized Lagrange interpolation collocation method (SLICM). The results show that the accuracy of the collocation methods based on the Lagrange interpolation function is higher than that of the collocation method based on the reproducing kernel function, and the accuracy and stability of the stabilized Lagrange interpolation collocation method are superior to those of the Lagrange interpolation collocation method.