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A novel two-grid method for semilinear elliptic equations. (English) Zbl 0795.65077
Author’s summary: 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 signficantly more expensive than the solution of one single linearized equation.

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N55Multigrid methods; domain decomposition (BVP of PDE)
35J65Nonlinear boundary value problems for linear elliptic equations