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
\[ -\nabla \cdot (\mathbf{a} \nabla u )+ \mathbf{b} \cdot \nabla u +cu=f \quad \text{in} \quad \Omega\subset \mathbb{R}^{2} \]
\[ u=0 \qquad \text{on} \quad \partial \Omega. \]
\(\Omega \) is a convex bounded convex polygonal domain and 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 \[ -\nabla \cdot (A(u) \nabla u )=f. \]
\[ -\nabla \cdot (\mathbf{a} \nabla u )+ \mathbf{b} \cdot \nabla u +cu=f \quad \text{in} \quad \Omega\subset \mathbb{R}^{2} \]
\[ u=0 \qquad \text{on} \quad \partial \Omega. \]
\(\Omega \) is a convex bounded convex polygonal domain and 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 \[ -\nabla \cdot (A(u) \nabla u )=f. \]
Reviewer: Erwin Schechter (Moers)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
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