Mixed Finite Element Methods For Several Classes Of Evolution Equations

An H~1-Galerkin mixed finite element method is discussed for two classes of second or-der equations: Schr(?)dinger equation and pseudo-hyperbolic integro-differential equation.Weseparate the real and imaginary part of the equation according the feature of the prob-lem.The H~1-Galerkin mixed finite method is studied for the real and imaginary part re-spectively.The optimal error estimates of the semi-discrete and fully discrete schemes ofthe Schr(?)dinger equation is derived by considering the two part of the equation simulta-neously.For the pseudo-hyperbolic integro-differential equation,depending on the physicalquantities of interest,two methods are discussed .Optimal error estimates of the function andits gradient are derived for semi-discrete scheme for problems in one space dimension, andthe extension to problems in two and three space variables also discussed.It is showed thatthe H~1-Galerkin mixed finite element approximations have the same rate of convergence asin the classical methods without requiring the LBB consistency condition.We construct the mixed discontinuous space-time finite element method for the integro-differential equations of the fourth order parabolic type .Lower the order of the equationby mixed finite element.And discretize the equation by space-time finite element method,contiuous in space but discontinuous in time. The stability, uniqueness and existence ,errorestimates of the approximate solution are proved.