computational efficiency of mixed least squares meshless models over sph method for elliptic pdes

Analysis of various physical phenomena requires a solution of partial differential equations. numerical methods especially the meshless schemes are used to discretize and solve these equations. recently, several modifications have been proposed and applied to improve the numerical performance of the smoothed particle hydrodynamics (sph) as a well-known meshless method. this study aims to show that the least squares methods especially in the context of the mixed formulation can get more accuracy compared with all existing sph models. to validate this claim, three meshless numerical models are derived from taylor series expansion on the basis of the mixed formulation. one of the models is based on first-order ordinary mixed formulation while two others use the first and second-order least squares mixed scheme. for accuracy analysis of the proposed methods, a potential flow and three 2d differential equations have been solved and examined with the presented sph models. the main finding of this study is that the proposed mixed second-order least squares method has the ability to achieve significant computational efficiency over several sph models. in addition, it has been found that the least square scheme can improve the accuracy of the ordinary mixed model.