Finite Volume Element Method With P₁ Nonconforming Element For Nonselfadjoint And Indefinite Elliptic Problems

In this paper,we consider the finite volume element method with the P₁ noncon-forming element for the second order non-selfadjoint and indefinite elliptic problems: whereΩ(?)R² is a bounded convex polygonal domain.As an auxiliary tool,we introduce the finite element method for the correspond-ing problem and derive the optimal order error estimates in the H¹-and L²-norms for the finite volume element approximations for the non-selfadjoint and indefinite elliptic problems,and obtain superconvergence estimates in the H¹-norm between the solution of the finite volume element method and that of the finite element method:With these superconvergence estimates,we establish the L^∞error estimate for the finite volume element method for the second order non-selfadjoint and indefinite elliptic problems:Moreover,based on two P₁ nonconforming finite element spaces S_H and S_h on one coarse grid with grid size H and one fine grid with grid size h,we consider the two-grid finite volume element discretization techniques for the non-selfadjoint and indefinite linear elliptic problems and the nonlinear elliptic problems.With the proposed techniques,solving the non-selfadjoint and indefinite problem on the fine space is reduced to solving a symmetric and positive definite problems on the fine space and the non-selfadjoint and indefinite elliptic problems on a much smaller space.If h=O(H²)is chosen,we show that the convergence rate of the two-grid methods is optimal in the H¹-norm: where u_h and u^h is the solution of the finite volume elemnet method and the solution of the two-grid method,respectively.