Low-order Raviart–Thomas approximations of axisymmetric Darcy flow

Abstract

Abstract In this paper, we study the lowest-order mixed finite element method for the axisymmetric Darcy problem using Raviart–Thomas elements. In contrast to the Cartesian setting, the method is non-conforming in the sense that the discretely divergence-free functions are not solenoidal. We derive several estimates that measure the inconsistency of the method and derive error estimates of the discrete pressure and velocity solutions. We show that if the domain is convex, then the errors converge with optimal order modulo logarithmic terms. Numerical experiments are presented, and they indicate that the estimates are sharp.