Solving One-dimensional Elliptic Interface Problems With Variable Coefficient By HDG Method

In this paper,an elliptic two-point boundary value problem with variable coefficients is taken as an example,and a numerical algorithm is proposed to solve the general interface problem with high accuracy.The basic idea is to construct a new quasi-function near the interface and use the function to con-vert the original problem equivalently into an extended interface problem with the interface of the mesh or grid.Due to the existence of the discontinuity coefficient,the quasi-function constructed on the unit near the interface does not explicitly know its analytic expression,but is coupled with the solution of the extended interface problem by a simple Hermite polynomial interpolation.It is worth noting that the use of coupling ideas does not affect the existence and uniqueness of the piecewise polynomial function and its third-order ap-proximation accuracy,thus ensuring that the extended interface problem has high regularity.Then,the hybridizable discontinuous Galerkin method is used to solve the extended interface problem.By selecting the numerical flux rea-sonably,the jump at the unit node is naturally introduced into the numerical format.The numerical example verifies the global matrix positive definite of the linear system of equations obtained by the method,and its condition num-ber is O(h-2).At the same time,the numerical examples in this paper show that the method is not only stable,but also has the second order convergence precision in the sense of L2 norm and L? norm.