Study of the convergence of the finite-element method for parabolic equations with a nonlinear nonlocal spatial operator
This paper is interested in a parabolic equation with a nonlinear non-local spatial operator. The considered spatial operator depends nonlinearly on the unknown function, its gradient, but also on a non-local (integral) spatial characteristic of the solution. The problem is semi-discretized with respect to the time \(t\). Then the finite element method is used for space discretization. The approximate solution is sought with the non-local characteristic resolved from the previous time level. The convergence of the constructed algorithm under minimal assumptions on the smoothness of the original data is proven.
scientific article; zbMATH DE number 6493786