This paper discusses the hybrid mixed finite element method for elliptic interface problem in which the solution and gradient are discontinuous because of jump conditions across the interface. For the twodimensional elliptic interface problem, we propose the hybrid mixed finite element method on triangulation. The properties of this method mainly reflect in the following: the triangular mesh can nicely approximate the complex interface|they provide the approximations of all the variables, including the solution and gradient, with the optimal order|the matrix of final linear algebra systems is symmetric and positive definite, so it can be numerically solved by using for example conjugate gradient method. Numerical examples demonstrate the validity of hybrid mixed finite element method for elliptic interface problems. 

The direct discontinuous Galerkin (DG) method was applied to calculate the soil instable temperature field of underground heated pipeline. This method applied partial integration from each computational cell for the heat transfer equation and obtained weak formulation.The direct DG was obtained by building appropriate numerical flux. The ordinary differential equations were gained by solving spatial discretization by means of the explicit Euler method. Through the calculation of the radial temperature field of heated pipeline ,its results coincide with other references'. Thus means that the direct discontinuous Galerkin method is an efficient and accurate numerical method for the calculation of instable temperature field.