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Two-grid finite volume element method for linear and nonlinear elliptic problems. (English) Zbl 1134.65077

First the finite volume element method (FVEM) is applied to solve the two-dimensional problem

-·(𝐚u)+𝐛·u+cu=finΩ 2
u=0onΩ·

Ω is a convex bounded convex polygonal domain and a symmetric and positive definite. The idea of the two-grid method is to reduce the non-selfadjoint and indefinite elliptic problem on a fine grid into a symmetric and positive definite elliptic problem on a fine grid by solving a non-selfadjoint and indefinite elliptic problem on a coarse grid.

In the last section the authors consider the FVEM for the two-dimensional second-order nonlinear elliptic problem with homogeneous boundary condition for

-·(A(u)u)=f·

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N15Error bounds (BVP of PDE)
65N55Multigrid methods; domain decomposition (BVP of PDE)
35J25Second order elliptic equations, boundary value problems
35J65Nonlinear boundary value problems for linear elliptic equations
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