Higher Accuracy Analysis For Nonlinear Advection-Diffusion Equation And Strongly Damped Wave Equations

This dissertation mainly includes two parts. In the first part, the optimal ε uniform convergent results of bilinear conforming finite element and nonconforming finite element EQ1rot are obtained under L2(Ω) norm for the time-dependent nonlinear advection-diffusion equations. Based on Bramble-Hilbert lemma, higher accuracy integral identities and some new asymptotic expansions are derived. Moreover, we derive the related approximate solutions with third order by use of the extrapolation technique. In the second part, the superconvergence analysis of H1-Galerkin mixed finite element method for strongly damped wave equations is studied. By virtue of the interpolation technique operator instead of Ritz projection of the original variable u and Ritz-Volterra projection of the stress variable p, the superclose and superconvergence results in in(Q) norm for u and H (div;Ω) norm for p for both semidiscrete and fully discrete schemes are derived through applying some error estimates of conforming linear triangular finite element.