(A) study on error estimates and an adaptive scheme of least-squares meshless method

Abstract

The least-squares meshless methods are presented where the shape functions are constructed by the moving least-squares (MLS) approximation and the variational equations are derived by the least-squares methods. It is shown that the least-squares formulations are highly robust to the integration errors while the Galerkin formulations are very sensitive to the integration errors. To investigate the mathematical structure concerned with the integration errors, inner products which reflect the integration errors are newly defined. Mathematical analyses show that the orthogonality of the solution error with respect to the finite dimensional subspace is reserved in the newly defined inner product in the least-squares methods while the orthogonality condition is not satisfied in the Galerkin methods. Numerical examples are also presented to verify that the least-squares methods are highly robust to the integration errors. Therefore the numerical integration can be performed with great ease and effectiveness using very simple algorithms. For the general and the elliptic first-order least-squares problems, \aposteriori error estimates are derived, respectively. The error indicator for a general problem is given as the square-root of the integral of the squared-residual in the influence domain of each node. For an elliptic problem, the error indicator can be improved by applying the Aubin-Nitsche method. It is demonstrated, through numerical examples, the error indicators reflect the actual error well. A simple nodal refinement scheme is presented making use of the Voronoi cells. For each node to be refined, new nodes are inserted at the vertices of the Voronoi cell of the node. Numerical examples show that the adaptive first-order least-squares meshless method is effectively applied to the localized problems such as the shock formation in fluid dynamics. The decoupled first-order least-squares meshless method and the second-order least-squares meshless method for t...