A Novel Adaptive Finite Volume Method for Elliptic Equations

Abstract

In this paper, we propose a novel adaptive finite volume method (AFVM) for elliptic equations. As a standard adaptive method, a loop of our method involves four steps: Solve $\rightarrow$ Estimate $\rightarrow$ Mark $\rightarrow$ Refine. The novelty of our method is that we do not have the traditional "completion" procedure in the Refine step. To guarantee the conformity, a triangular element with a hanging node is treated as a quadrilateral element, and the corresponding function space consists of the bilinear functions. The optimal computational complexity of our AFVM is validated by numerical examples.