A Class of Finite Volume Schemes of Arbitrary Order on Nonuniform Meshes

作者：Zhang, Qinghui; Zou, Qingsong^*

来源：Numerical Methods for Partial Differential Equations, 2014, 30(5): 1614-1632.

DOI：10.1002/num.21853

• high order
• finite volume method
• singular solution
• non-uniform partition

In this article, we generalize the bi-k,k1 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 H-1 norm of our finite volume schemes' discretization error converges with optimal order O(N-k), while the L-2 norm error converges with order O(N-(k 1)). Here, N-2 is the cardinality of the underlying mesh. Superconvergence property of the scheme is also discussed. Our theoretic findings have been verified by numerical experiments.

• 出版日期2014-9
• 单位中山大学