An H~1-Galerkin Expanded Mixed Finite Element Simulation For Second Order Nonlinear Evolution Equations

Many practical phenomenons,such as nonlinear(linear) wave problems,acous-tic problems,environmental fluid mechanics,are governed by nonlinear(linear)evo-lution equations,so the numerical simulation for such problems become a hot spot in applied mathematics,computational mathematics and engineering science.Sec-ond order hyperbolic equation is a special kind of evolution equations,which de-scribes such important phenomenons as nonlinear(linear)wave and acoustic prob-lems.Based on the special concerns on the sound pressure (the unknown scalar), the gradient of sound pressure(the gradient of unknown scalar)and the acceleration of sound transmission(the adjoint vector)in engineering application, in this paper we propose an H1-Galerkin expanded mixed finite element method for second order hyperbolic equations and intend to approximate the desired three quantities simul-taneously with high accuracy. The numerical analysis verify that this new method is an ideal numerical method for this equation.We first discuss the following second order linear hyperbolic initial-boundary value problem which is used to govern linear wave. An H1-Galerkin expanded mixed formulation is proposed,the equivalence of the weak form and the initial-boundary value problem is proved,and the existence and uniqueness of discrete solutions is given.And then the optimal L2 estimates of three variables are derived.This shows this new method have both merits of the expanded mixed finite element method and H1-Galerkin method,that is,the optimal estimates for the unknown scalar,the gradient of unknown scalar and the adjoint vector,being free of LBB stability condition and the finite element spaces having different polynomial degrees.Another advantage of this method we find so far is no need to invert the coefficient tensor, which assures the applicability for problem with small coefficient.In fact,the most of the sound transmissions are in nonlinear form,so we further discuss the following nonlinear hyperbolic problem An H1-Galerkin expanded mixed formulation is proposed,the equivalence between the weak form and the initial-boundary value problem is proved,and the existence and uniqueness of discrete solutions is given.And then the optimal L2 estimates of three variables are derived by an inductive hypothesis.The theoretical analysis show that the method is a high-performance method for the nonlinear wave equa-tion.Also we find that the new method is no need to invert the coefficient tensor and can simulate the transmission in high density media, and this formulation over-comes the difficulties resulted from the differentiation of a(p)with respect to time t as done in H1-Galerkin mixed finite element method.