Numerical Analysis Of Sobolev Equation And Purely Longtudinal Motion Equation Of A Homogeneous Bar

In this paper, the mixed finite element method for approximate Sobolev equation and the finite volume element method for approximate purely longtudinal motion equation of a homogeneous bar are considered. The optimal( or quasi-optimal) order error estimates for the two kinds of schemes are derived.In Chapter one, The object is to investigate the convergence of the mixed finite element method of the initial-boundary value problem of Sobolev equationbased on the Raviart-Thomas space V_h × W_h (?) H(div;Ω) × L~2(Ω). Optimal order estimates are obtained for the approximation of u in L~∞(0, T;L~2(Ω)) and the associated velocity p in L~∞(0,T;L~2(Ω)~2),divp in L~∞(0,T;L~2(Ω)). Quasi-optimal order estimates are obtained for the approximation of u in L~∞(0, T;L~∞(Ω)) when the index k = 0 and p in L~∞(0,T;L~∞(Ω)~2). Optimal order estimate is also obtained for the approximation of u in L~∞(0,T;L~∞(Ω)) when k ≥ 1.In Chapter two, we consider the finite volume method for approximate the initial-boundary-value problem of purely longtudinal motion equation of ahomogeneous bar(a) utt = uxxt + f(ux)x, (x,t) e (0,1) x [0, T),(b) u(x, 0) = uo(x), ut(x: 0) = 1*1(3;), x e (0,1),(c) ?(0,*) = u(M) = 0, te [0,T].We obtain optimal order error estimates in L2 and H1, superconvergence error estimates in H1 and give a numerical experimint of this scheme.