Staggered Taylor-Hood and Fortin elements for Stokes equations of pressure boundary conditions in Lipschitz domain.

*(English)*Zbl 07258861Summary: The purpose of this paper is to develop a general theory on how the inf-sup stable and convergent elements of the velocity Dirichlet boundary (VDB)-Stokes problem with no-slip VDB are still inf-sup stable and convergent for the pressure Dirichlet boundary (PDB)-Stokes problem with PDB in Lipschitz domain. The PDB-Stokes problem in a Lipschitz domain usually only has a singular velocity solution which does not belong to \((H^1(\Omega))^2\), sharply in contrast to the VDB-Stokes problem whose velocity solution still belongs to \((H^1(\Omega))^2\), and unexpectedly, some well-known inf-sup stable and convergent VDB-Stokes elements may or may no longer correctly converge. It turns out that the inf-sup condition of the PDB-Stokes problem in Lipschitz domain relies on an unusual variational problem and requires adequate degrees of freedom on the domain boundary. In this paper we propose two families of staggered elements: staggered Taylor-Hood elements \((\mathcal{CP}_{\ell+2})^2 - \mathcal{P}_\ell\) with \(\ell \geq 1\) (continuous in both velocity and pressure) and staggered Fortin elements \((\mathcal{CP}_{m+2})^2 - \mathcal{P}_m^{\text{disc}}\) with \(m \geq 1\) (continuous in velocity and discontinuous in pressure) on triangles, for solving the PDB-Stokes problem in Lipschitz domain. We show that the two families are inf-sup stable and are correctly convergent for the non-\(H^1\) singular velocity. Numerical results illustrate the proposed elements and the theoretical results.

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

76M10 | Finite element methods applied to problems in fluid mechanics |

35Q35 | PDEs in connection with fluid mechanics |