Unsteady Hyperbolic Problem Discontinuous Finite Element Method

Finite Element Method is the modern science and engineering in computing the one of the most important numerical method. Since the1960s, Finite Element Method is applied successfully in the solving elliptical and parabolic issues, but a hyperbolic problems and to order is not very effective. The main difficulties in two respects:The first is a order of the solution of the problem (in particular the nonsmooth problems) there may be hyperbolic physical discontinuous(Weak discontinuity or Strong discontinuity), by ordinary solved by finite element method will produce the physical oscillation when, can’t accurately simulate the physical process; The second is a first order hyperbolic equation has no elliptic or parabolic equation variational problem of positive definite and boundedness of the structure, the theoretical analysis is difficult, the general finite element methods are difficult to achieve the optimal order of convergence. In order to overcome these difficulties, people on the hyperbolic problem introduced various new forms of the finite element method. For example: Characteristic and Upwind Finite Element Method, Finite Volume Element Method and Discontinuous Finite Element Method and so on.Discontinuous Finite Element Methods are the traditional finite element method (continuous) form of innovation, improvement and development. Discontinuous Finite Element Method used in the unit at the junction of completely continuous piecewise polynomial space as the trial and test function space of a class of finite element method,it is widely applied to many practical fields, such as meteorology, oceanography, gas dynamics, fluid mechanics, oil exploration and other. Discontinuous Finite Element Method for lifting of the ordinary (coordination) finite element method across the boundary continuity constraints, it is this that the discontinuous finite element method has many good properties, mainly include:the physical conservation properties in the unit to meet, good stability, especially suitable for smooth not high hyperbolic problem, is a high resolution numerical method.This paper based on the basic principle of discontinuous finite element method, presents two classic discontinuous finite element method, namely intended to windward of Discontinuous Finite Element Method and forms of punishment Discontinuous Finite Element Method. According to the basic principle of finite element, introduces a kind of can solve one explicit space and time unit discontinuous finite-element method, structural unsteady hyperbolic equations first explicit space and time format discontinuous finite element method, and gives stability estimate.The method that keeping the finite element of high precision and greatly reduces the amount of calculation. In particular, discussed a solve unsteady linear convection dominant diffusion problem discontinuities streamline spread the finite element method, when diffusion coefficient and division of the grid parameters than proper hours, this paper gives the format of the stability analysis and error order estimate.