A finite volume approximations for one nonlinear and nonlocal integrodifferential equations

In this paper, the error analysis of the Petrov–Galerkin finite volume element method (FVEM) is investigated for a nonlinear parabolic integro-differential equation that arises in the mathematical modeling of the penetration of a magnetic field into a substance, accounting for temperature-dependent changes in electrical conductivity. Starting from Maxwell’s equations, we derive a one-dimensional model problem, which forms the basis of our analysis. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. For simplicity, we consider only the lowest-order (linear and L-splines) finite volume elements. The novel contribution lies in the application of FVEM to this problem, leading to the establishment of an unconditionally stable numerical scheme and the derivation of optimal error estimates in the $L^{\infty }\left(L^2\left(\Omega \right)\right)$ and $L^2\left(H_0^1\left(\Omega \right)\right)$ norms for both semi-discrete and linearized backward Euler fully-discrete schemes, using a generalized projection method that carefully manages the nonlinear terms. Lastly, numerical experiments are provided to support the theoretical conclusions.