Two-grid Methods for Finite Volume Element Approximations of Nonlinear Sobolev Equations

In this article, two-grid methods are studied for solving nonlinear Sobolev equation using the finite volume element method. The methods are based on one coarse grid space and one fine grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H), and the fine grid solution (with grid size h) can be obtained in a single symmetric and linear step. The optimal H¹ error estimates are presented for the proposed methods, which show that the two-grid methods achieve optimal approximation as long as the mesh sizes satisfy h = 𝒪(H³|ln H|). As a result, solving such a large class of nonlinear Sobolev equations will not be much more difficult than solving one linearized equation.