Numerical Methods For Solving The Two-Point Elliptic Interface Problems

In this paper,taking the two-point elliptic interface problem as an example,based on the theory of reproducing kernel space and Legendre polynomial,and the properties of linear operators,two numerical algorithms with high accuracy are obtained.In the first method,a new Hilbert space is constructed firstly using(2₂⁸)space,and it is proved that it is a reproducing kernel space,and the reproducing kernel function of the space is given.Then,a group of basis functions of the reproducing kernel space is constructed by using the property of the reproducing kernel function,and the approximate solution space is obtained.Therefore,we can use a linear representation of the basis function to represent the approximate solution.In order to obtain the approximate solution of the equation and prove that the unknown coefficients are uniquely determinable,an algebraic system of linear equations’s matrix form is considered.Finally,the error analysis of this method is also discussed.In the second method,firstly,according to the Legendre polynomials,a set of basis functions of the reproducing kernel space(2₉)[(6,(7]are constructed,and the reproducing kernel functions are obtained.Accordingly,a new reproducing kernel space₉)[0,1]is constructed.The corresponding reproducing kernel function⁸)（）is obtained,and then the approximate solution of the model equation is obtained by using the method mentioned above.Moreover,the existence and uniqueness of the solution is proved.