Superconvergence Of Splitting Positive Definite Mixed Finite Element Methods For Hyperbolic Equations-the Continuous-time Case

There are many works about the superconvergence of partial di?erential equa-tions. At present, there have existed many numerical methods to solve the secondorder hyperbolic equations. Among these numerical methods, mixed finite elementmethod plays a very important role. Now, a splitting positive definite mixed finiteelement method have been established for solving partial di?erential equations, themethod avoids of the saddle point problems which leads from the classical mixedmethod.Scientific researchers have done many works on the convergence property ofsplitting positive definite mixed finite element method for partial di?erential equa-tions. Danping Yang obtained the convergence property of splitting positive defi-nite mixed finite element method for miscible displacement of compressible ?ow inporous media in [46]. Then, this methods is applicable to second order hyperbolicequations by Jiansong Zhang , see [51]. He established a splitting positive definitemixed finite element procedure for second order hyperbolic equations and obtainedthe convergence of the mixed element method with continuous- and discrete-timescheme.In this paper, We apply the splitting positive definite mixed finite elementmethods for hyperbolic equations, and obtain a mixed weak form and a new mixedvariational form of hyperbolic equations. Then , we consider the superconvergenceproperty of a semi-discrete scheme splitting positive definite mixed finite elementmethod for second order hyperbolic problems. We introduce the elliptic projectionfollow [12, 6, 47]. it is easy to obtain the superconvergence between the ellipticprojection and Raviart-Thomas projection. So the important results of this paperis to obtain the superconvergence between the mixed finite element solution andthe elliptic projection.