Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number
arXiv:1703.00156
Abstract
We study superconvergence property of the linear finite element method with the polynomial preserving recovery (PPR) and Richardson extrapolation for the two dimensional Helmholtz equation. The -error estimate with explicit dependence on the wave number {is} derived. First, we prove that under the assumption ( is the mesh size) and certain mesh condition, the estimate between the finite element solution and the linear interpolation of the exact solution is superconvergent under the -seminorm, although the pollution error still exists. Second, we prove a similar result for the recovered gradient by PPR and found that the PPR can only improve the interpolation error and has no effect on the pollution error. Furthermore, we estimate the error between the finite element gradient and recovered gradient and discovered that the pollution error is canceled between these two quantities. Finally, we apply the Richardson extrapolation to recovered gradient and demonstrate numerically that PPR combined with the Richardson extrapolation can reduce the interpolation and pollution errors simultaneously, and therefore, leads to an asymptotically exact {\it a posteriori} error estimator. All theoretical findings are verified by numerical tests.
arXiv comment25 pages, 16 figures. arXiv admin note: substantial text overlap with arXiv:1612.03386