Discontinuous Finite Element Methods For Nonlinear Differential Equations

Recently, reserches about discontinuous finite element methods are veryhot.In this paper, we introduce some kinds of discontinuous finite elementmethods and their formulations. Especialy, we focus on discontinuous finiteelement methods for nonlinear ordinary differential equations and quasi-linearhyperbolic equations.We study even degree(κ≥0) discontinuous finite element method fornonlinear ordinary differential initial value problems. Constructing a new localdiscrete Green function, using Legendre orthogonal expansion in an elementto construct a comparative function and using dual argument and continuousmethod, we prove that numerical flux at nodes (?)_j=(U_j^-+U_j⁺)/2 have thehighest order superconvergence O(h^2k+2). This highest superconvergence orderis confirmed by our numerical experiments and superconvergence points inelements are also observed in our error figures. In addition, we also discuss thenumber of superconvergence points in one element.Reserches on quasi-linear hyperbolic problems with discontinuous solu-tion are very difficult, most of the methods are differential methods.We applya discontinuous finite element method of time-space full discretization to com-pute the Riemann Problem. Our numerical experiments show that there is nooscillation and it simulates rarefaction wave and shock wave well.