DG Method For Solving Linear And Nonlinear Differential Equations

In this paper, discontinuous Galerkin method (DG) is discussed to solve the two-dimensional singularly perturbed problem and semilinear differential equations. For the latter, the so-called interpolated coefficient finite element method is combined to deal with the nonlinear term.As we calculated the approximate solutions of singularly perturbed convection-diffusion equations in the one-dimensional setting, we find that the supercon-vergence of DG method is stronger than that of traditional finite element method at nodes. Moreover, the DG method also can simulate the acute variation of solution[29]. Inspired by those results, we use the DG method to solve two-dimensional singularly perturbed problem. Under uniform mesh and two kinds of Shishikin mesh, our numerical results obtained by DG method verify the super convergence and no oscillation is observed.With regard to the multiple solutions of semilinear elliptic equations, DG method is efficient to deal with the sharp peak of solutions, which will occur when the morse index is high, in the domain or on the boundary. Our numerical results are also presented to show the efficiency of the method.