Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data
The paper deals with a parabolic integro-differential equation of the form \[ u_t + A(t)u = \int_0^t B(t,s) u(s) ds + f \tag{1} \] in \(\Omega \times (0,T]\) subjected to the boundary and initial conditions \[ u = 0 \quad \text{on} \quad \partial \Omega \times (0,T], \qquad u(\cdot,0) = u_0 \quad \text{in} \quad \Omega. \tag{2} \] Here \(\Omega \subset R^d\) is a bounded domain with smooth boundary, \(T < \infty\). The coefficients of the second-order linear partial differential operators \(A(t), B(t,s)\) are smooth, moreover \(A(t)\) is supposed to be a selfadjoint and uniformly positive definite elliptic operator. Integro-differential equations of the above form occur in many applications as heat conduction in materials with memory, compression of poroviscoelastic media, and nuclear reactor dynamics. The paper was stimulated by an open problem posed by \textit{C. Chen} and \textit{T. Shih} [Finite element methods for integro-differential equations, Singapore (1998; Zbl 0909.65142)]. Introducing a weak formulation of (1), (2) the authors consider its finite element Galerkin approximation with respect to the space variable (with continuous time \(t \in (0,T]\)). First, a priori estimates for the exact solution are obtained, using standard energy methods and a suitably defined Ritz-Volterra projection. In the sequel the error estimates are proved in the following two cases: (i) \(f=0\), \(u_0 \in H^2(\Omega)\) or \(u_0 \in L^2(\Omega)\) and (ii) \(f(t) \in L^2\Omega)\), \(u_0 = 0\).
parabolic integro-differential equations
heat conduction in materials with memory
scientific article; zbMATH DE number 1884270