, Volume 46, Issue 2, pp 293-304

On error estimates in the Galerkin method for hyperbolic equations

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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.

Original Russian Text Copyright © 2005 Zhelezovski \(\overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{\imath}\) S. E.
Translated from Sibirski \(\overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{\imath}\) Matematicheski \(\overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{\imath}\) Zhurnal, Vol. 46, No. 2, pp. 374–389, March–April, 2005.
This revised version was published online in April 2005 with a correction of misspelled author names.