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We investigate anH1-Galerkin expanded mixed finite element approximation of nonlinear second-order hyperbolic equations, which model a wide variety of phenomena that involve wave motion or convective transport process. This method possesses some features such as approximating the unknown scalar, its gradient, and the flux function simultaneously, the finite element space being free of LBB condition, and avoiding the difficulties arising from calculating the inverse of coefficient tensor. The existence and uniqueness of the numerical solution are discussed. Optimal-order error estimates for this method are proved without introducing curl operator. A numerical example is also given to illustrate the theoretical findings.

We investigate aH1-Galerkin mixed finite element method for nonlinear viscoelasticity equations based onH1-Galerkin method and expanded mixed element method. The existence and uniqueness of solutions to the numerical scheme are proved. A priori error estimation is derived for the unknown function, the gradient function, and the flux.