Pressure-robust and conforming discretization of the Stokes equations on anisotropic meshes
Volker Kempf
TL;DR
The paper addresses the challenge that conventional Stokes discretizations can lose velocity accuracy when pressure gradients drive the solution. It proposes a conforming, pressure-robust discretization based on a reconstruction operator I_h applied to the Bernardi–Raugel element, enabling anisotropic meshes to be used without sacrificing robustness. By incorporating a Helmholtz–Hodge decomposition of the forcing, preserving discrete divergence, and proving stability and consistency results, the authors establish a priori error bounds: ||u−u_h||_{1,h} ≤ C ( inf_{v_h∈X_h^0} ||u−v_h||_{1,h} + h ||P(Δu)||_0 ) and ||p−p_h||_0 ≤ C ( inf_{q_h∈Q_h} ||p−q_h||_0 + (ν/β̃) inf_{v_h∈X_h^0} ||u−v_h||_{1,h} + (h/β̃) ||f||_0 ). Numerical experiments on boundary-layer problems with anisotropic Shishkin-type meshes demonstrate a pronounced velocity-robust advantage and highlight the influence of mesh anisotropy on both velocity and pressure errors. The proposed BR-RT and BR-BDM variants provide a practical, conforming route to pressure-robust Stokes simulations in challenging anisotropic settings.
Abstract
Pressure-robust discretizations for incompressible flows have been in the focus of research for the past years. Many publications construct exactly divergence-free methods or use a reconstruction approach [13] for existing methods like the Crouzeix--Raviart element in order to achieve pressure-robustness. To the best of our knowledge, except for our recent publications [3,4], all those articles impose a condition on the shape-regularity of the mesh, and the two mentioned papers that allow for anisotropic elements use a non-conforming velocity approximation. Based on the classical Bernardi--Raugel element we provide a conforming pressure-robust discretization using the reconstruction approach on anisotropic meshes. Numerical examples support the theory.