Legendre Galerkin Chebyshev Collocation Least Squares Method For Two Kinds Of Linear Differential Equations With Variable Coefficients

The Legendre Galerkin(LG)collocation least squares spectral method is studied for variable coefficient elliptic equation and parabolic equation.For the elliptic equation with variable coefficients,the LG collocation least squares method is given.Combined with the Chebyshev collocation method,a discrete system with symmetric positive definite is obtained,which is easy to solve by iterative method;In numerical analysis,this method is coupled with the good stability of Legendre method;In numerical calculations,due to the fast calculation characteristics of Chebyshev interpolation,it makes the processing of variable coefficients more advantageous.And by selecting a reasonable basis function,the stiffness matrix and mass matrix obtained are sparse.The given example verifies the high-order accuracy of the numerical solution method.The LG collocation least squares spectral method is applied to Darcy flow problems,in which homogeneous and heterogeneous media are considered respectively,and the results are compared with those of the corresponding numerical methods.Numerical experiments show the feasibility and high accuracy of the method.In addition,for the time-dependent model,the least squares method of the parabolic equation is considered.Based on the Legendre Galerkin form,the time direction is discretized by the difference method,and the space direction is based on the Legendre method.The initial value and source term are processed by the LGL or CGL configuration method.The numerical experimental results given show the effectiveness of the method.For the parabolic equation with variable coefficients,the numerical format of the least squares of the space-time LG spectrum configuration is also given.For the parabolic equation with variable coefficients,the LG spectral collocation time-space least squares method is also given...