On error estimates in the Galerkin method for hyperbolic equations

Abstract

We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form.Keywords: second-order hyperbolic equation, the Galerkin method, error estimate, weak solution, existence and uniqueness of a solution, finite element method

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

“…The proof of this theorem is carried out mainly by the available methods; therefore, here we only sketch the main steps. The assertion on existence of a unique weak solution to (0.1), (0.2) is a consequence of the corresponding result of [1] (which is proven under somewhat more general conditions on B as compared with those of this article). The proof of unique strong solvability of (1.7), (1.8) is carried out by the scheme of [11, Chapter 1, Sections 1.4-1.6] combined with some technical elements of [1].…”

“…The assertion on existence of a unique weak solution to (0.1), (0.2) is a consequence of the corresponding result of [1] (which is proven under somewhat more general conditions on B as compared with those of this article). The proof of unique strong solvability of (1.7), (1.8) is carried out by the scheme of [11, Chapter 1, Sections 1.4-1.6] combined with some technical elements of [1]. For construction of a strong solution to (1.7), (1.8), we use semidiscrete Galerkin approximations and establish a priori estimates of the form (1.11) and (1.12) for them; their proof relies upon the Bihari lemma [18, p. 189, Theorem 3] and the Gronwall lemma [19, p. 191], respectively.…”

“…The obtained estimate (see (2.1) below) guarantees strong convergence of approximate solutions to the exact solution in the energy norm for an arbitrary choice of the sequence of the projection subspaces which is limit-dense in V . A similar error estimate for the semidiscrete Galerkin method (without discretization in time) was established in [1]. Apparently, the error estimate of this article is new as well as the convergence of the projection-difference method in the energy norm for hyperbolic, in particular, second-order linear equations in such general situation when the projection subspaces are arbitrary, the free term satisfies the only condition f ∈ L 1 (0, T ; H) and hence the problem has only a weak exact solution.…”

“…The properties of the function f ε needed below can be found in [1]. Alongside the initial problem (0.1), (0.2) and scheme (1.4), (1.5), we consider the problem…”

“…We will use the definition of a weak solution to (0.1), (0.2) as introduced in [1]. moreover, if τ is sufficiently small (τ ≤ τ 0 , where τ 0 ∈]0, T 0 /2] is a constant independent of n and ε) then we also have the estimate…”