Meshless Methods Theoretical Research In Partial Differential Equations

Now, meshfree method is an active topic of the recent research in the areas of computational science and approximation theory. As is it named, the methods used a kind of functions approximation which are just based on a finite number of nodes instead of both nodes and meshes as in finite element method. When using it to solve partial difference equations, the meshes can be completely or partly eliminated. This is an significant advantage compared with finite element method which is one of outstanding achievements made in the side of computation methods last century. Over the past decade meshfree approximation methods have found their way into many different application areas ranging from artifcial intelligence, computer graphics, image processing and optimization to the numerical solution of all kinds of (partial) differential equations problems.In the thesis, firstly we have clarified the background knowledge of meshfree methods, and educed the meshfree methods after introducing the key ideas and disadvantage of finite element method. Secondly three kinds of approaches of function approximation are discussed which are the interpolation with radial basis function, moving least squares and approximate approximation. And the correlative theories are proved. Then some details about the two kinds of meshfree methods for numerical solution of partial differential equations, which are element free Galerkin method (EFGM) and meshless local Petrov-Galerkin (MLPG) method, are analysised. After comparing the integral schemes of EFGM with that of MLPG method through both abundant numerical examples and computer cost analysis, we conclude that the background cell integral method used by EFGM originally can save a lot of computer cost compared with the method of integration on local supporting areas used by MLPG method originally under the same precision. A modified Lagrange mutipler methods through indentifying the multipler is posed to achieve theessential boundary conditions. And some comparison results between it and the other methods such as Lagrange multipler methods and facter methods are given through numerical examples. All of the three methods has their own characters of both sides. Finally the both sufficient and necessary condition of the selection of radius of influence in moving least squares is posed and researched further. To get better approximation, the best value of radius of influence is explored by enough numerical experimentation. When the nodes are distributed uniformly, we suggest that the best radius of influence can be chosen as 1.2h or 2.2h (where h is step) as p(x) are chosen as linear bases or quadratic bases respectively.