The Discontinuous Galerkin Finite Element Method For Elliptic Equation

Reed and Hill frst proposed discontinuous Galerkin fnite element method (referredto DG fnite element method) in1973. Discontinuous Galerkin methods are a class offnite element methods using completely discontinuous basis functions. Their fexibilityand benefts are not available in traditional fnite element methods, such as the allowanceof arbitrary triangulation and hanging nodes. So the DG fnite element methods havebeen rapid development and applications in many felds.At present, there are two modes for solving elliptic equations with DG fnite elementmethod: one is solving the difusion equation of the DG fnite element method applied toelliptic problems, namely the introduction of intermediate variables, forming frst orderequations; the other is direct application of a form of punishment, including symmetricinterior penalty Galerkin (SIPG) nonsymmetric interior penalty Galerkin (NIPG) andincomplete interior penalty Galerkin (IIPG) methods. In two modes, the numerical fuxand the interior penalty are respectively used to suppress discontinuous across elementboundary.In this paper, the DG methods are applied to solving one-dimensional and two-dimensional elliptic problems. First of all, we apply three mixed discontinuous fniteelement methods to two points boundary value problem with mixed boundary conditions.And we extend to the case of discontinuous coefcients. Secondly, symmetric interiorpenalty Galerkin (SIPG) nonsymmetric interior penalty Galerkin (NIPG) and incom-plete interior penalty Galerkin (IIPG) methods are used to solve elliptic equation withvariable coefcients. From the view of the convergence order and the penalties' value,above three methods are compared. We conclude the advantages and disadvantages ofthem. Finally, we propose the new NIPG and IIPG methods to achieve the optimalconvergence order.