In this paper, we study low-order mixed finite element approximations of three-dimensional Darcy flow in axisymmetric domains and with axisymmetric data. Due to the symmetry of the problem, the original three-dimensional problem is reduced to a two-dimensional one, and therefore this reduction offers considerable less computational effort to approximate the solutions. On the other hand, the axisymmetric formulation necessitates the use of weighted function spaces and modified (singular) differential operators leading to theoretical difficulties. This is especially true for mixed finite element methods because their structure-preserving properties often do not hold in the axisymmetric setting.
We derive error estimates of the direct mixed finite element method using the lowest-order RT elements. The main difficulty in the analysis is that, in contrast to the Cartesian setting, the axisymmetric divergence operator acting on the RT space is not surjective onto the space of piecewise constants. Therefore, in this sense, the method is non-conforming.
The approach we take in the analysis is classical. We simply apply Strang's second lemma to obtain abstract error estimates in terms of the approximation properties of the RT space and the inconsistency (or non-conformity) of the method. We then derive several estimates of the (local) inconsistency of the method, each tailored for different regions of the domain. Using a convex combination of these estimates, and by applying a dyadic decomposition of the domain with respect to the r-variable, we show that the inconsistency of the method is almost first-order provided that the domain is convex.