A NOVEL 2-GRID METHOD FOR SEMILINEAR ELLIPTIC-EQUATIONS

A new finite element discretization technique based on two (coarse and fine) subspaces is presented for a semilinear elliptic boundary value problem. The solution of a nonlinear system on the fine space is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse space and a linear system on the fine space. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation. As a result, the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation.