In this work we introduce and analyse a new adaptive PetrovGalerkin heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations. In a general heterogeneous setting we prove convergence of the HMM approximations to the solution of a macroscopic limit equation. The major new contribution of this work is an a-posteriori error estimate for the L2-err...

The Moving Least Squares method (MLS) provides an approximation û of a function u based solely on values u(xj) of u on scattered ”meshless” nodes xj . Derivatives of u are usually approximated by derivatives of û. In contrast to this, we directly estimate derivatives of u from the data, without any detour via derivatives of û. This is a generalized Moving Least Squares technique, and we prove t...

The finite element method (FEM) has been commonly employed in a variety of fields as a computer simulation method to solve such problems as solid, fluid, electro-magnetic phenomena and so on. However, creation of a quality mesh for the problem domain is a prerequisite when using FEM, which becomes a major part of the cost of a simulation. It is natural that the concept of meshless method has ev...

The general Meshless Local Petrov-Galerkin (MLPG) type weak-forms of the displacement & traction boundary integral equations are presented, for solids undergoing small deformations. Using the directly derived non-hyper singular integral equations for displacement gradients, simple and straight-forward derivations of weakly singular traction BIE's for solids undergoing small deformations are als...

This paper presents a new approach based on the meshless local Petrov–Galerkin (MLPG) and collocation methods to treat the parabolic partial differential equations with non-classical boundary conditions. In the presented method, the MLPG method is applied to the interior nodes while the meshless collocation method is applied to the nodes on the boundaries, and so the Dirichlet boundary conditio...

For the first time in mathematical finance field, we propose the local weak form meshless methods for option pricing; especially in this paper we select and analysis two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpolation (LRPI) based on Wu’s compactly supported radial basis functions (WCS-RB...

The application of the local meshless numerical method (LRBFCM) for solving a system of coupled partial differential equations (PDE) is explored. The numerical approach is tested on the natural convection based fluid flow problems. The fluid flow part of the solution procedure is coupled locally despite its global nature. Such an approach makes the computations convenient for an implementation ...