Two-grid finite volume element method for linear and nonlinear elliptic problems

Authors:
Chunjia Bi;Victor Ginting
Affiliations:
Yantai University, Department of Mathematics, 264005, Shandong, People’s Republic of China;Colorado State University, Department of Mathematics, 80523, Fort Collins, CO, USA
Venue:
Numerische Mathematik
Year:
2007

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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 two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional 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 efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates.