On the Crank-Nicolson anisotropic a posteriori error analysis for parabolic integro-differential equations (Q2814444)

The authors consider an inhomogeneous linear parabolic integro-differential equation with a second-order self-adjoint positive definite elliptic differential operator and a Volterra integral operator with kernel \( - \nabla\cdot(\mathcal{B}(s,t)\nabla u)\). The problem is discretized using linear finite elements in space and the Crank-Nicolson scheme in time. The finite element meshes are allowed to be anisotropic, i.e., the aspect ratio of the elements need not to be bounded, which is a frequent assumption made in the analysis but limiting the applicability in analyzing finite element subdivisions often used in effective calculations.NEWLINENEWLINEThe paper starts from a paper by \textit{A. Lozinski} et al. [SIAM J. Sci. Comput. 31, No. 4, 2757--2783 (2009; Zbl 1215.65154)] on error estimators for parabolic equations without memory term using anisotropic elements. The space error estimates are obtained with the aid of methods from the article by \textit{L. Formaggia} and \textit{S. Perotto} [Numer. Math. 94, No. 1, 67--92 (2003; Zbl 1031.65123)]. For estimating the error due to time discretization piecewise quadratic polynomial interpolation is used. The kernel of the memory term is linearly approximated. The time derivative of the nonlocal memory term is approximated using a three-point difference approximation of \(u_{tt}\) which, by virtue of the given integro-differential equation, differs from the memory term by local quantities only. Some assumptions on the geometry of the meshes are imposed.NEWLINENEWLINEThe result of the procedure described above is condensed in an optimal order a posteriori error estimate. The statement is quite complex consuming a whole print page. A similar second theorem is derived using a different approximation of the memory term. Numerical studies are not reported.