Semidiscrete and single step fully discrete approximations for second order hyperbolic equations

Abstract

Finite element approximations are analysed, for initial boundary value problems far second order hypebolic equations. For both semidiscrete and fully discrete schemas, optimal order rate o f convergence estimates in L2 are derived, using L2 projections of the initial data as starting values. A new class of single step fully discrete schemes is developed, which are high order accurate in time. The schemes are constructed from a class of rational approximations to e^-z, analytic in neighbourhoods of the imaginary axis. The approximations require the solution of 2s linear systems at each time step, with the same real matrix, to yield convergence rate k^s, where k is the time step and s is an arbitrary positive integer.