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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 $L^\infty$-norm for finite element approximations to the parabolic integro-differential equation $$u_t+V(t)u(t)=f(t),\quad t\in (0,T); \qquad u(0)=u_0,$$ where $V(t)$ denotes a general (Volterra) integro-differential operator on a Hilbert space; typically, $$V(t)u(t)= A(t)u(t)+ \int_0^t B(t,\tau)u(\tau)d\tau,$$ with $A(t)$ linear and elliptic of second order, and $B(t,\tau)$ is a linear differential operator of order not exceeding two. The derivation of sharp $L^\infty$ 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 {\it Y. P. Lin, V. Thomée} and {\it 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.

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