$P_1$-Nonconforming Quadrilateral Finite Volume Element Method and Its Cascadic Multigrid Algorithm for Elliptic Problems

Abstract

In this paper, we discuss the finite volume element method of $P_1$-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic multigrid algorithm is proposed to solve the nonsymmetric large-scale system resulting from such discretization. Numerical experiments are reported to support our theoretical results.