A general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well-posed linear operator equations is presented. The results provide optimal convergence rates, if the test and trial spaces satisfy a stability condition. Operators need not be elliptic, and the problems can be posed in weak or strong form without changing the t...

This contribution concerns the construction of meshless Lagrangian particle methods for the numerical simulation of multiscale phenomena in linear transport problems, where mass conservative discretization methods are essentially required. The proposed discretization scheme works with a finite set of unstructured nodes, each corresponding to one flow particle. In this method, corresponding part...

A particular meshless method, named meshless local Petrov–Galerkin is investigated. To treat the essential boundary condition problem, an alternative approach is proposed. The basic idea is to merge the best features of two different methods of shape function generation: the moving least squares (MLS) and the radial basis functions with polynomial terms (RBFp). Whereas the MLS has lower computa...

In this paper the meshless local Petrov-Galerkin (MLPG) method is implemented to study the buckling of isotropic cylindrical shells under axial load. Displacement field equations, based on Donnell and first order shear deformation theory, are taken into consideration. The set of governing equations of motion are numerically solved by the MLPG method in which according to a semi-inverse method, ...

The truly meshless local Petrov-Galerkin (MLPG) method holds a great promise in solving boundary value problems, using a local symmetric weak form as a natural approach. In the present paper, in the context of MLPG and the meshless interpolation of a moving least squares (MLS) type, a method which uses primary and secondary nodes in the domain and on the global boundary is introduced, in order ...

We present a new meshless method for scalar diffusion equations, which is motivated by their compatible discretizations on primal-dial grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points xi. This graph defines a local primal-dual grid complex with a virtual dual grid, in the sense that sp...

A comparison study of the efficiency and accuracy of a variety of meshless trial and test functions is presented in this paper, based on the general concept of the meshless local Petrov-Galerkin (MLPG) method. 5 types of trial functions, and 6 types of test functions are explored. Different test functions result in different MLPG methods, and six such MLPG methods are presented in this paper. I...

The numerical implementation of the truly Meshless Local Petrov-Galerkin (MLPG) type weakforms of the displacement and traction boundary integral equations is presented, for solids undergoing small deformations. In the accompanying part I of this paper, the general MLPG/BIE weak-forms were presented [Atluri, Han and Shen (2003)]. The MLPG weak forms provide the most general basis for the numeri...

Abstract. The simplest pair of spaces P1/P0 is made inf-sup stable for the mixed form of the Darcy equation. The key ingredient is enhance the finite element spaces inside a PetrovGalerkin framework with functions satisfying element-wise local Darcy problems with right hand sides depending on the residuals over elements and edges. The enriched method is symmetric, locally mass conservative and ...