Dispersion and pollution of the improved meshless weighted least-square (IMWLS) solution for the Helmholtz equation

The meshless weighted least-square (MWLS) method is a meshless method based on the moving least-square (MLS) approximation. Compared with the Galerkin based meshless methods, the MWLS avoids numerical integrations, which improves the computational efficiency significantly. The MLS may form ill-conditioned system of equations, an accurate solution of which is difficult to obtain. In this paper, by using the weighted orthogonal basis function to construct the improved moving least-square (IMLS) approximation and the Lagrange multiplier method to enforce the Dirichlet boundary condition, we derive the formulas and perform the dispersion analysis for an improved meshless weighted least-square (IMWLS) method for two-dimensional (2D) Helmholtz problems. Results demonstrated that the IMWLS is more accurate and has advantages in handling dispersion. A 2D industrial model problem illustrated that the proposed method can easily reach higher frequency without losing accuracy.