## 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