Discrete Fluxes and Mass Balance in Finite Elements

Significant mass balance "errors" have been reported with finite-element techniques, leading to sweeping statements that "Finite elements do not conserve mass locally." These reports of mass balance "error" emerge when mass flux estimates are based upon post-processing of the approximated solution variables. Fluxes computed in this way appear to be nonphysical. For example, even under steady-state conditions, the flux entering and leaving a closed region may appear unequal. This has serious consequences for a typical model application that might require the estimation of flow division in a branching stream, or details about the effects of sources or sinks. In this paper, we show that by remaining consistent with the discrete approximation given by the finite-element statement, the resulting flux estimates will preserve mass balance. As a consequence, it can be shown that general finite-element approaches for hydraulic engineering do, indeed, conserve mass locally, as well as globally. These locally conservative, "consistent" flux estimates are computed for time-dependent mass conservation in one and three dimensions.