Superconvergent and Divergence-Free Mixed Finite Element Methods for The Stokes Equation

Abstract

This paper develops divergence-free mixed finite element methods for the Stokes equation. Using H(div)-conforming velocities and discontinuous pressures ensures the inf-sup condition for the velocity--pressure pair and yields pointwise divergence-free velocities. However, this choice makes the vector Laplacian difficult to discretize. Inspired by mass-conserving mixed formulations with stresses, tangential--normal continuous traceless tensor elements are used to discretize the vector Laplacian. An inf-sup condition for the weak div operator between the stress and velocity spaces is then proved. Two key properties characterize the scheme. First, the stress--velocity inf-sup stability gives a stable discretization of the vector Laplacian without additional stabilization, unlike discontinuous Galerkin or virtual element methods. Second, the scheme has the property that if a stress field is distributionally divergence-free against the discrete divergence-free velocity space, then it is also distributionally divergence-free against the continuous divergence-free velocity space. This property decouples the stress and velocity errors and leads to superconvergence. As a result, optimal-order error estimates are obtained for the stress, while the discrete velocity is superclose to its H(div) interpolant. The projected-pressure error estimate is optimal with Raviart-Thomas velocities and superconvergent with Brezzi-Douglas-Marini velocities, while local postprocessing yields an elementwise divergence-free velocity with higher-order convergence. Numerical experiments confirm the theoretical results.