We analyze finite volume schemes of arbitrary order r for the one-dimensional singu- larly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as (N-11n(N...We analyze finite volume schemes of arbitrary order r for the one-dimensional singu- larly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as (N-11n(N 4- 1))r, where 2N is the number of subinter- vals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order (N-11n(N + 1))2r, while at the Gauss points, the derivative error super-converges with order (N-11n(N + 1))r+1. All the above conver- gence and superconvergence properties are independent of the perturbation parameter e. Numerical results are presented to support our theoretical findings.展开更多
文摘We analyze finite volume schemes of arbitrary order r for the one-dimensional singu- larly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as (N-11n(N 4- 1))r, where 2N is the number of subinter- vals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order (N-11n(N + 1))2r, while at the Gauss points, the derivative error super-converges with order (N-11n(N + 1))r+1. All the above conver- gence and superconvergence properties are independent of the perturbation parameter e. Numerical results are presented to support our theoretical findings.