Two-grid methods for a new mixed finite element approximation of semilinear parabolic integro-differential equations

In this paper, we present a two-grid scheme for a semilinear parabolic integro-differential equation using a new mixed finite element method. The gradient for the method belongs to the space of square integrable functions instead of the classical $H(\mathrm{div};\Omega)$ space. The velocity and the pressure are approximated by a $P_0^2-P_1$ pair which satisfies an inf-sup condition. Firstly, we solve the original nonlinear problem on the coarse grid in our two-grid scheme. Then, to linearize the discretized equations, we use Newton’s iteration on the fine grid twice. It is shown that the algorithm can achieve an asymptotically optimal approximation as long as the mesh sizes satisfy $h=\mathcal{O}(H^6|\ln H|^2)$. As a result, solving such a large class of nonlinear equations will not be much more difficult than solving one linearized equation. Finally, a numerical experiment is provided to verify the theoretical results of the two-grid method.