A class of finite volume schemes of arbitrary order on nonuniform meshes1. Issue 5 (23rd December 2013) - Zhang, Qinghui; Zou, Qingsong
Abstract In this article, we generalize the bi‐<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghvbtpd0d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />k, k≥1</alternatives> 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 <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghvbtpczd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />r2|logr|</alternatives>, 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 <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghvbtpd2f" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />O(N−k)</alternatives>, while the L2 norm error converges with order <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghvbtpd1x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />O(N−(k+1))</alternatives>. 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