Superconvergence Of Full-discrete Spdmfe For Second-order Hyperbolic Equations

Since 1970s, people have fruitful achievements in the analysis of hyperbolic equations. Optimal rates of convergence for continuous or discrete time Galerkin finite element methods are analyzed for linear or nonlinear second order hyper-bolic equations in several spaces. The convergence for the classical mixed element methods of the hyperbolic equations is presented, too. However, the technique of the classical mixed method leads to some saddle point problems whose numerical solutions have been quite difficult because of losing positive definite properties.In this paper, we analyze splitting positive definite mixed element methods for second order hyperbolic equations with a series of superconvergence theory. We can prove the superconvergence of the error estimates which is O(h2) between the exact solution and its finite element approximation. First, we use a new mixed element method to solve the second order hyperbolic equation, in which the coefficient matrix of the mixed element system is symmetric positive definite. Second, the discrete-time mixed scheme will be defined and analyzed. Finally, the global superconvergence is obtained by the superconvergence theory and the interpolated post-processing technique.Note:In the topic, "SPDMFE"is abbreviated from Splitting Positive Definite Mixed Finite Element.