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Zbl 1134.65077
Bi, Chunjia; Ginting, Victor
Two-grid finite volume element method for linear and nonlinear elliptic problems. (English)
[J] Numer. Math. 108, No. 2, 177-198 (2007). ISSN 0029-599X; ISSN 0945-3245/e

First the finite volume element method (FVEM) is applied to solve the two-dimensional problem $$-\nabla \cdot (\bold{a} \nabla u )+ \bold{b} \cdot \nabla u +cu=f \quad \text {in} \quad \Omega\subset \mathbb{R}^{2}$$ $$u=0 \qquad \text {on} \quad \partial \Omega.$$ $\Omega $ is a convex bounded convex polygonal domain and {\bf 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 $$ -\nabla \cdot (A(u) \nabla u )=f. $$
[Erwin Schechter (Moers)]

MSC 2000:: *65N30 Finite numerical methods (BVP of PDE)
65N15 Error bounds (BVP of PDE)
65N55 Multigrid methods; domain decomposition (BVP of PDE)
35J25 Second order elliptic equations, boundary value problems
35J65 (Nonlinear) BVP for (non)linear elliptic equations

Keywords: finite volume element method; two-grid method; second-order nonlinear elliptic problem

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