A Taylor-Hood finite element method for the surface Stokes problem without penalization

TL;DR

This work addresses high-order finite element approximation of the surface Stokes problem on a smooth surface $\gamma$, where enforcing tangential velocity and $H^1$-conformity simultaneously is challenging. It introduces a tangential, penalty-free Taylor--Hood SFEM by constructing the velocity space $\boldsymbol{V}_h$ via Piola transforms with nodal DOFs anchored to local tangent planes and by mapping the pressure space $Q_h$ to degree $r-1$ polynomials, with edge DOFs placed at Gauss--Lobatto points. The authors prove discrete inf-sup stability, derive geometric consistency estimates, and establish optimal-order convergence in the energy norm and velocity $L^2$ norm, including an $L^2$ velocity estimate via a duality argument. Numerical experiments corroborate that Gauss--Lobatto placement of edge DOFs is essential for optimal convergence and demonstrate that the method extends the Demlow–Neilan approach to higher-order Taylor--Hood discretizations on isoparametric geometries.

Abstract

Finite element approximation of the velocity-pressure formulation of the surfaces Stokes equations is challenging because it is typically not possible to enforce both tangentiality and $H^1$ conformity of the velocity field. Most previous works concerning finite element methods (FEMs) for these equations thus have weakly enforced one of these two constraints by penalization or a Lagrange multiplier formulation. Recently in [A tangential and penalty-free finite element method for the surface Stokes problem, SINUM 62(1):248-272, 2024], the authors constructed a surface Stokes FEM based on the MINI element which is tangentiality conforming and $H^1$ nonconforming, but possesses sufficient weak continuity properties to circumvent the need for penalization. The key to this method is construction of velocity degrees of freedom lying on element edges and vertices using an auxiliary Piola transform. In this work we extend this methodology to construct Taylor-Hood surface FEMs. The resulting method is shown to achieve optimal-order convergence when the edge degrees of freedom for the velocity space are placed at Gauss-Lobatto nodes. Numerical experiments confirm that this nonstandard placement of nodes is necessary to achieve optimal convergence orders.

Figures (4)

• Figure 1: A visual description of constructing tangent planes at each node via the operator $\mathcal{M}_a^K$. In theconstruction of the finite element space, we take ${\bm \nu}_{\rm ref} = {\bm \nu}_{K_a}(a)$.
• Figure 2: Convergence in the energy norm $|\breve \boldsymbol{u}-\boldsymbol{u}_h|_{H_h^1(\Gamma_{h,k})}+ \|p^e-p_h\|_{L_2(\Gamma_{h,k})}$ (left) and $L_2$ norm $\|\breve \boldsymbol{u} -\boldsymbol{u}_h\|_{L_2(\Gamma_{h,k})}$ (right) for quadratic, cubic, and quartic Taylor-Hood elements using Gauss-Lobatto edge degrees of freedom.
• Figure 3: Convergence in the energy norm $|\breve \boldsymbol{u}-\boldsymbol{u}_h|_{H_h^1(\Gamma_{h,k})}+ \|p^e-p_h\|_{L_2(\Gamma_{h,k})}$ (left) and $L_2$ norm $\|\breve \boldsymbol{u} -\boldsymbol{u}_h\|_{L_2(\Gamma_{h,k})}$ (right) for quadratic, cubic, and quartic Taylor-Hood elements using Lagrange degrees of freedom.
• Figure 4: Convergence in the energy norm $|\breve \boldsymbol{u}-\boldsymbol{u}_h|_{H_h^1(\Gamma_{h,k})}+ \|p^e-p_h\|_{L_2(\Gamma_{h,k})}$ and $L_2$ norm $\|\breve \boldsymbol{u} -\boldsymbol{u}_h\|_{L_2(\Gamma_{h,k})}$ for cubic Taylor-Hood elements using Lagrange degrees of freedom, and cubic and quartic surface approximations.

Theorems & Definitions (48)

• Remark 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Remark 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Definition 3.1
• Remark 3.2