Collocated Mixed Discrete Least Squares Meshless (CMDLSM) method for solving quadratic partial differential equations

Abstract

In this paper, a Collocated Mixed Discrete Least Squares Meshfree (MDLSM) method is proposed and used for efficient solution of engineering problems. Background mesh is not required in the MDLSM method; hence the method is a truly meshfree method. Nodal points are used in the MDLSM methods to construct the shape functions while collocated points are used to form the least squares functional. In the original MDLSM method, the location of the nodal points and collocated points are the same. In the proposed Collocated Mixed Discrete Least Squares Meshfree (CMDLSM) method, a set of additional collocated points is introduced. It is expected that the accuracy of results is improved by using the additional collocated points. It is noted that the size of coefficient matrix is not increased in the proposed CMDLSM method compared with the MDLSM method. The required computational effort for solving the linear algebraic system of equations, therefore, is the same as that of MDLSM method. A set of benchmark numerical examples, cited in the literature, is used to evaluate the performance of proposed method. The results indicate that the accuracy of solutions is improved by using additional collocated points in the proposed CMDLSM method.