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Two-Grid Finite Volume Element Method for Linear and Nonlinear Elliptic Problems

Abstract. Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the second-order non-selfadjoint and indefinite linear elliptic problems and the second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the eficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates.

Mathematics Subject Classication (2000). 65N15 65N30