Superconvergent and Divergence-Free Mixed Finite Element Methods for The Stokes Equation
This work addresses a computational bottleneck in fluid dynamics simulations for researchers and engineers, offering an incremental improvement over existing methods like discontinuous Galerkin or virtual element methods.
The paper tackled the challenge of discretizing the vector Laplacian in divergence-free mixed finite element methods for the Stokes equation by introducing tangential–normal continuous traceless tensor elements, resulting in optimal-order error estimates for stress and superconvergence for velocity and pressure.
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 weakly divergence-free, then it is also strongly divergence-free. This 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.