Any order superconvergence finite volume schemes for 1D general elliptic equations
arXiv:1207.0566
Abstract
We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy and norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special case, the convergence rate can reach , where is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.
24 pages, 6 figures