Analysis of a Surface Crouzeix-Raviart Element for the Stokes Problem

• Surface Stokes equations model incompressible viscous flows constrained to Earth’s spherical geometry, motivating robust, affordable discretizations for climate models.Source: §1 · p. 1
• The method uses a tangential Crouzeix-Raviart velocity space on a polyhedral surface and piecewise constant pressure, but its velocity space is H1(Γh)-nonconforming.Its edge-midpoint construction uses a continuous tangent vector and a conormal vector that is discontinuous along edges.Source: §1 · p. 2
• Because the symmetric strain-rate operator has a non-trivial kernel on the nonconforming space, the formulation stabilizes the momentum equation to address velocity oscillations.The stabilization includes an edge-jump penalty term.Source: §1 · p. 2
• The analysis establishes discrete inf-sup stability and derives energy- and L2-error estimates, with geometric, pressure-coupling, and nonconformity terms requiring careful control.The proposed formulation is presented as a computationally efficient, easy-to-implement alternative to conforming and H(divΓ)-based approaches.Source: §1 · pp. 2–3

• The exact surface Γ is an oriented, connected, bounded, smooth surface without boundary, and a closest-point projection maps a sufficiently small tubular neighborhood onto Γ.Source: §2 · p. 3
• Scalar functions are extended from Γ along its normal direction and lifted between the exact and discrete surfaces; vector-valued functions use these operations component-wise.Source: §2 · pp. 3–4
• The preliminaries define the projection onto the tangent plane and use it to set out surface differential operators, including covariant derivative and surface divergence.Source: §2 · p. 4
• The notation specifies L2 and L∞ norms, zero-mean square-integrable functions, and scalar and vector-valued Sobolev spaces, including tangential vector fields.Source: §2 · p. 4

• The surface Stokes problem seeks a tangential velocity field and scalar pressure, with the momentum equation involving the tangential strain-rate tensor.Source: §3 · p. 5
• Because the surface is closed, no boundary conditions are imposed, and a zeroth-order velocity term is included to avoid technical difficulties from the surface strain operator’s non-trivial kernel.Source: §3 · p. 5
• For a smooth, compact, closed surface, prior results establish coercivity and the continuous inf-sup condition, and the weak problem is well-posed for the stated data.Source: §3 · p. 5

4.1 Surface approximation and associated results

• The surface is approximated by planar triangles whose vertices lie on the exact surface, and elements and edge directions are related to the curved surface through lifting.The discrete and lifted tangential and conormal vectors need not align, so geometric errors arise.Source: §4.1 · pp. 6–7

4.2.1 Tangential vector-valued Crouzeix-Raviart space

• The velocity space is constructed from tangential Crouzeix–Raviart functions whose tangential and conormal components are continuous across edge midpoints.The pressure space consists of piecewise-constant functions.Source: §4.2.1 · pp. 9–10

4.2.2 Discrete formulation

• The discrete formulation defines a strain-rate tensor and energy norm, and its coercivity relies on a discrete Korn inequality.The text identifies verification of the discrete inf-sup condition as the remaining step for establishing existence and uniqueness.Source: §4.2.2 · pp. 10–11

4.2.3 Discrete inf-sup stability result

• The discrete inf-sup condition holds with a constant independent of mesh size for the Crouzeix–Raviart velocity and piecewise-constant pressure spaces.The proof uses interpolation properties that preserve conormal jumps and divergence under lifting.Source: §4.2.1 · p. 10, §4.2.3 · pp. 12–14

5.1 Interpolation estimates

• The interpolation subsection establishes velocity interpolation bounds and introduces an L2-projection operator for pressure with a corresponding interpolation estimate.The estimates combine velocity interpolation results with a pressure interpolation estimate recalled from.Source: §5.1 · pp. 14–15

5.2 Energy-error estimation

• The energy-error analysis applies the Second Strang Lemma to bound the error through approximation and nonconformity contributions.Coercivity and discrete inf-sup stability also imply a unique discrete solution; the nonconformity analysis controls geometric-error terms and jumps.Source: §5.2 · p. 16

5.3 L2-error estimate

• The L2-error analysis uses Aubin–Nitsche duality, combining primal and dual consistency, interpolation, pressure-coupling, and nonconformity estimates.A displayed bound is ≤ Ch^2∥g∥L2(Γ)∥f∥L2(Γ).Source: §5.3 · pp. 20, 21, 23, 24
• Combining the estimates for error terms I11–I21 with Ph(ue − uh) = ge yields (88).The geometric terms use Lemmas 7, 13, and 14; the primal-dual terms use Cauchy–Schwarz, (80), and the energy estimates (68) and (75), with Lemmas 16 and 18 bounding the remaining terms.Source: §5.3 · p. 25

6 Numerical experiment

• The experiments use exact solutions on a unit sphere and a torus with major radius 1 and minor radius 0.5, measuring broken H1 and L2 velocity errors and L2 pressure error.The study tests three stabilization parameters on both surfaces.Source: §6 · pp. 26–27
• On both the sphere and torus, the errors show the predicted first-order H1 velocity and pressure convergence and second-order projected L2 velocity convergence.Figure 3 plots the error measures against mesh size for both surfaces.Source: Figure 3 · p. 28, §6.1 · p. 29, §6.2 · pp. 29–30

6.1 Results on the Sphere

• On the sphere, H1 velocity errors are smallest at α = 0.5, projected L2 velocity errors are relatively insensitive to α, and pressure L2 error decreases as α increases.The experiments use α = 0.5, 1.0, and 2.0.Source: §6 · p. 27, §6.1 · p. 29

6.2 Results on the Torus

• On the torus, H1 velocity and pressure errors decrease as α increases, with the smallest errors at α = 2.0, while projected L2 velocity error remains largely insensitive to α.The torus results retain the predicted convergence rates for all three tested stabilization parameters.Source: §6.2 · pp. 29–30