Accepted

Superconvergent and Divergence-Free Finite Element Methods for Stokes Equation

Long Chen, Xuehai Huang, Chao Zhang, and Xinyue Zhao

SIAM Journal on Numerical Analysis

arXiv   Bibtex

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
introduced 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
velocity and pressure converge at rates higher than the approximation
orders of the chosen spaces. Numerical experiments confirm the
theoretical results.