Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients

Abstract

<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm error estimates are proved for finite element approximations to the solutions of initial boundary value problems for second order hyperbolic partial differential equations with time-dependent coefficients. Optimal order rates of convergence are shown for semidiscrete and single step fully discrete schemes using specially constructed initial data. The initial data are designed so that the data used for the fully discrete equation is reasonable to compute and so that the optimal order estimates can be proved.