H～1-Galerkin Mixed Finite Element Method For Hyperbolic Equation And Its Numerical Analysis

In the first part of this paper, with fully-discrete H~1-Galerkin mixed finite element method, we consider the second-order linear hyperbolic partial differential equationWe give two fully-discrete H~1-Galerkin mixed finite element schemes in one space and several spaces. One method discretes the time derivative directly and obtains a three-level scheme; Another method turns the equation into the parabolic system through the conventional transformation and obtains a two level-scheme. We obtain the approximates of the scalar unknown and its flux (the coefficient times the gradient) optimally and stimultaneously. The approximating finite element spaces V_h and W_h are allowed to be of differing polynomial degrees for the proposed method. Moreover, this method does not need to satisfy the LBB consistancy condition. We obtain the optimal order of convergence theoretically.Then we consider the nonlinear hyperbolic problem andwhich is stimulated by semi-discrete HI-Galerkin mixed finite element method. Through numerical analysis we prove the optimal L~2-norm and H~1-norm error estimates for approximating the unknown function, the adjoint vector fuction as well as the time derivative of the adjoint vector function.