Lin, Y. P. On maximum norm estimates for Ritz-Volterra projection with applications to some time dependent problems. (English) Zbl 0879.65097 J. Comput. Math. 15, No. 2, 159-178 (1997). 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 Y. P. Lin, V. ThomĂ©e 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. Reviewer: H.Brunner (St.John’s) Cited in 12 Documents MSC: 65R20 Numerical methods for integral equations 45K05 Integro-partial differential equations 45N05 Abstract integral equations, integral equations in abstract spaces Keywords:Ritz-Volterra projection; stability; error estimates; finite element; parabolic integro-differential equation; Hilbert space Citations:Zbl 0728.65117 PDFBibTeX XMLCite \textit{Y. P. Lin}, J. Comput. Math. 15, No. 2, 159--178 (1997; Zbl 0879.65097)