Two-grid Methods for Characteristic Finite Volume Element Approximations of Semi-linear Sobolev Equations

In this paper, two -grid methods for characteristic finite volume element solutions are presented for the semi linear Sobolev equation. The method is based on the methods of characteristics, two -grid method and the finite volume element method. 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 by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two -grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = O(H-3). As a result, solving such a large semi -linear Sobolev equations will not be much more difficult than solving one single linearized equation.