Research On RBF-FD Method For Elliptic Equation With Variable Coefficient

In actual theoretical research,mathematical models(such as elliptic partial differential equations)are usually used to describe physics and engineering problems,such as diffusion problems,quantum mechanics problems,electromagnetic problems,and current distribution problems in conductors,geometry,fluid dynamics,and electrostatics.However,the boundary value problem of this kind of equation can only obtain its exact solution under some special conditions,so it is very important to study the numerical method of these problems.The finite difference method is a numerical solution method for elliptic equations with a relatively wide range of application and good results.Until now,there are many difference methods that can be used to approximate elliptic equations.For example,direct difference method and finite volume method can be used to deal with one-dimensional variable coefficient elliptic equations.However,these methods have lower convergence accuracy.It can only reach the second order.To improve the accuracy of the difference method,the grid needs to be encrypted,but this will greatly increase the computational burden.Many difference methods can also be used to solve the two-dimensional equation boundary value problem,but the accuracy is generally low.Therefore,to solve the boundary value problem of the elliptic equation with variable coefficients well,it is of great theoretical and practical significance to establish a difference scheme with high accuracy and good convergence.Here are the main contents of this article:(1)The Lagrangian form of the radial basis function is used to derive the radial basis function expansion formula,and any order finite difference of the interpolation function at the nodes is given.The truncation error of the first order difference,and the best expression of the shape parameter ε contained in the RBF-FD formula is obtained.(2)Using the established RBF-FD formula to study the numerical solution methods of the first-type and two-dimensional variable-coefficient elliptic equations for the first kind of boundary condition problems.Three-point difference schemes for non-homogeneous two-point boundary value problems of one-dimensional variable-coefficient elliptic equations and nine-point difference schemes for two-dimensional variable-coefficient equations are established respectively.The overall format has fourth-order accuracy.The optimal value of the shape parameter is calculated,and the numerical experiment results prove that the difference format with the optimal parameter value is very effective and feasible.