On the Convergence of Galerkin Approximation Schemes for Second-Order Hyperbolic Equations in Energy and Negative Norms

Abstract

Abstract. Given certain semidiscrete and single step fully discrete Galerkin approximations to the solution of an initial-boundary value problem for a second-order hyperbolic equation, //' and L2 error estimates are obtained. These estimates are valid simultaneously when the approximation to the initial data is taken to be the projection onto the approximating space with respect to the inner product which induces the energy norm that is naturally associated with the problem. The L2-estimate is obtained as a by-product of the analysis of convergence in certain negative norms. Estimates are also obtained for the convergence of higher-order time derivatives in the presence of sufficiently smooth data.

Semidiscrete and single step fully discrete finite element approximations for second order hyperbolic equations with nonsmooth solutions

“…Ho wever, the results in [4] do not include the case when the initial data u° and u°t are in L 2 which is the case considered in this work.…”

“…In [4], Geveci proved energy and négative norm estimâtes for the solution of (1.1). Ho wever, the results in [4] do not include the case when the initial data u° and u°t are in L 2 which is the case considered in this work.…”

The convergence of a Galerkin approximation scheme for an extensible beam

“…As in baker and Bramble [1], Thomée [11] and Geveci [6], we will introducé another inner product on H 2 x L 2 :…”

“…Remark : From the proof it is clear that we also have which, in turn, implies where ||.||_ 2 dénotes the norm of the dual of H 2 , as in [6]. 3.…”

The convergence of Galerkin approximation schemes for second-order hyperbolic equations with dissipation

“…Then, with u(t) = Dtu(t), we have TD,ù(t) + u(t) + aTu(t) = 0, and parallel to the treatment in [1] and [5], an evolution equation for U(t) = [u(t),ù(t)]T:…”