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Zbl 0728.65117
Lin, Yanping; Thomée, Vidar; Wahlbin, Lars B.
Ritz-Volterra projections to finite-element spaces and applications to integrodifferential and related equations.
(English)
[J] SIAM J. Numer. Anal. 28, No.4, 1047-1070 (1991). ISSN 0036-1429; ISSN 1095-7170/e

The numerical solution of partial integodifferential equations (with homogeneous Dirichlet boundary conditions and given initial values) by time-continuous finite-element methods is considered. The paper presents convergence results based on the decomposition $u\sb h-u=(u\sb h-V\sb hu)+(V\sb hu-u)$ of the error, where $V\sb h$ is the so-called Ritz- Volterra projection. \par First, various error estimates for the Ritz-Volterra projection (in $L\sb p$ for $2\le p\le \infty)$ are given. Separate sections are then devoted to their application to parabolic and hyperbolic integrodifferential equations, and to Sobolev and viscoelasticity type equations.
[E.Hairer (Genève)]
MSC 2000:
*65R20 Integral equations (numerical methods)
45K05 Integro-partial differential equations

Keywords: Sobolev equation; time-continuous finite-element methods; convergence; Ritz-Volterra projection; parabolic; hyperbolic; viscoelasticity type equations

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