A class of finite volume schemes of arbitrary order on nonuniform meshes1. Issue 5 (23rd December 2013) - Zhang, Qinghui
Abstract In this article, we generalize the bi‐k, k≥1 finite volume schemes developed in Zhang and Zou, J Sci Comput (in press) for elliptic equations with high smooth solutions to elliptic equations with singular solutions. By designing a special nonuniform rectangular meshes, we construct a class of finite volume schemes of arbitrary order k. Our theoretic analysis shows that if the solution has weak singularity of type r2|logr|, where r is the distance from some target point to some fixed singular point, the H1 norm of our finite volume schemes' discretization error converges with optimal order O(N−k), while the L2 norm error converges with order O(N−(k+1)). Here, N2 is the cardinality of the underlying mesh. Superconvergence property of the scheme is also discussed. Our theoretic findings have been verified by numerical experiments. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1614–1632, 2014