Discontinuous Galerkin Method For Parabolic Problem Equation With Interior Penalty

The finite element method is a very effective numerical method to solve the partial differential equation, main points as below, firstly make the partial differential equation turn to equivalent variational problem, and use the finite dimensional space approximate the infinite dimensional space, then acquire the approximate solution-finite element solution in finite dimensional space. The finite element method theory has already established in the early60s, for traditional continuum element this theory has almost developed fully. The finite element method is applied to many fields, such as:manufacturing, medical, construction and aviation technology etc.Discontinuous Galerkin method which has advantage of traditional Galerkin method and finite volume element method become a hot topic in the field of scientific computing. Discontinuous Galerkin method remove the limit of boundary continuity, which makes the discontinuous Galerkin method has many good properties. This article mainly research on discontinuous Galerkin method of parabolic problem.The article start form discontinuous Galerkin method of elliptic problem, use method which is transition from finite element method of elliptic problem to parabolic problem, We can establish the discontinuous Galerkin method of parabolic problem. Firstly the article introduce finite element method of elliptic problem and parabolic problem, then discuss semi-discrete and fully discrete of parabolic problem, and derive stability of solution and error estimation of semi-discrete formulation separately and get error estimation of backward Euler discretization and Crank-Nicolson discretization of fully-discrete discontinuous finite element method.