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On maximum norm estimates for Ritz-Volterra projection with applications to some time dependent problems. (English) Zbl 0879.65097

The author studies the stability and pointwise error estimates in the L -norm for finite element approximations to the parabolic integro-differential equation

u t +V(t)u(t)=f(t),t(0,T);u(0)=u 0 ,

where V(t) denotes a general (Volterra) integro-differential operator on a Hilbert space; typically,

V(t)u(t)=A(t)u(t)+ 0 t B(t,τ)u(τ)dτ,

with A(t) linear and elliptic of second order, and B(t,τ) is a linear differential operator of order not exceeding two. The derivation of sharp L error estimates is based on a certain adjoint equation whose solution may be viewed as a regularized Green’s function associated with the Ritz-Volterra operator [compare an earlier paper by Y. P. Lin, V. Thomée and L. B. Wahlbin, SIAM J. Numer. Anal. 28, No. 4, 1047-1070 (1991; Zbl 0728.65117)]. The results are applied to a number of concrete problems: parabolic integro-differential equations, Sobolev’s equation, and a diffusion equation with a nonlocal boundary condition. There are no numerical examples.

MSC:
65R20Integral equations (numerical methods)
45K05Integro-partial differential equations
45N05Abstract integral equations, integral equations in abstract spaces