Higher Order Single Step Fully Discrete Approximations for Second Order Hyperbolic Equations with Time Dependent Coefficients

Abstract

Error estimates are proved for finite element approximations to the solution of an initial boundary value problem for a second order hyperbolic partial differential equation with time dependent coefficients. Optimal order rates of convergence in $L^2 $ are shown for single step fully discrete schemes. The order $\nu $ of the single step schemes is greater than or equal to one and less than or equal to four. A class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained. These methods avoid the repeated work of new matrix factorizations at each time level.