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.

Supplement to the Convergence of Galerkin Approximation Schemes for Second-Order Hyperbolic Equations With Dissipation

“…In the formulation and presentation of the convergence analysis, we shall mainly follow Geveci [5]. Let X denote the space X = H1^) X L2(Q) with the 'energy' inner product…”

“…The semidiscrete Galerkin approximation to (1.1) is derived from the formulation 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:…”

“…Our aim is to extend the convergence analysis for conservative hyperbolic equations by Geveci [5] and Baker and Bramble [1] to include a dissipative term of the above form.…”

“…The problem (1.1) and its approximation is considered in the framework of certain subspaces Hs(tl) of the Sobolev spaces Hs(il), as in [1], [2], [5] and [7]: HS(Q) = {v g Hs(Sl):LJv = 0 on 8fi for; < s/2), s > 0.…”

Study of convergence of the projection-difference method for hyperbolic equations

“…In the literature we can find numerous energy estimates for the error of the projection-difference method for second-order hyperbolic partial differential equations (for example, see [7][8][9][10]). However, all estimates of this kind, known to the author, do not apply or guarantee convergence of the method in the situation under study (the free term satisfies only the condition f ∈ L 1 (0, T ; H) and the problem has only a weak exact solution).…”