Nonconforming Finite Volume Methods for Second Order Elliptic Boundary Value Problems
The work provides a theoretical foundation for nonconforming FVMs, benefiting researchers in numerical analysis of elliptic problems, but the results are incremental extensions of existing finite element theory.
The paper develops a general convergence theorem for nonconforming finite volume methods (FVMs) using nonconforming finite element spaces, proving optimal error estimates for Crouzeix-Raviart and hybrid Wilson FVMs on regular meshes, with numerical validation.
This paper is devoted to analyze of nonconforming finite volume methods (FVMs), whose trial spaces are chosen as the nonconforming finite element (FE) spaces, for solving the second order elliptic boundary value problems. We formulate the nonconforming FVMs as special types of Petrov-Galerkin methods and develop a general convergence theorem, which serves as a guide for the analysis of the nonconforming FVMs. As special examples, we shall present the triangulation based Crouzeix-Raviart (C-R) FVM as well as the rectangle mesh based hybrid Wilson FVM. Their optimal error estimates in the mesh dependent $H^1$-norm will be obtained under the condition that the primary mesh is regular. For the hybrid Wilson FVM, we prove that it enjoys the same optimal error order in the $L^2$-norm as that of the Wilson FEM. Numerical experiments are also presented to confirm the theoretical results.