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A penalty free non-symmetric Nitsche type method for the weak imposition of boundary conditions

Erik Burman

arXiv:1106.5612v2math.NAcs.NA

TL;DR

For continuous approximation spaces, penalty-free stability was not established by the available DG analysis. This paper proves stability and optimal H1 convergence for the Poisson problem, with an L2 convergence rate that is suboptimal by half an order in h, and discusses convection–diffusion extensions.

  • Problem

    For continuous approximation spaces, the paper addresses the lack of established stability results for penalty-free non-symmetric Nitsche methods, since DG analysis does not directly apply under continuity constraints.

    Source: §1 · p. 1

  • Method

    The paper analyzes the non-symmetric Nitsche formulation with penalty parameter γ set to zero, using an inf-sup argument to recover boundary control.

    Source: §3 · p. 3, §4 · p. 4

  • Results

    For the Poisson problem, the method is stable and optimally convergent in H1 for k ≥1, while the L2 error rate is suboptimal by half an order in h.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The parameter-free method is presented as an addition to the computational toolbox, particularly for flow problems with non-symmetric system matrices, and is extended to convection–diffusion settings.

    Source: §1 · pp. 1–2

    The paper proves results for weak boundary imposition, and the boundary-penalty formulation yields the smallest L2 error, so the penalty-free method is not claimed to be the most accurate.

    Source: §1 · p. 2

Abstract

from arXiv · show

In this note we show that the non-symmetric version of the classical Nitsche's method for the weak imposition of boundary conditions is stable without penalty term. We prove optimal $H^1$-error estimates and $L^2$-estimates that are suboptimal with half an order in $h$. Both the pure diffusion and the convection--diffusion problems are discussed.

1. Introduction.

The paper addresses the unproven stability of penalty-free non-symmetric Nitsche’s method with continuous approximation spaces. It proves stability and optimal H1 convergence for the Poisson problem, while noting the L2 rate is suboptimal by half an order in h.

1. Introduction.

  • The classical symmetric Nitsche method uses a penalty parameter with a lower bound for coercivity; later non-symmetric DG analyses established penalty-free stability for polynomial orders k ≥2.The non-symmetric method was proposed by Freund and Stenberg, but omitting the penalty then caused coercivity failure and degenerating error estimates; Oden et al. proposed a DG version.Source: §1 · p. 1
  • For continuous approximation spaces, the authors identify a gap: the penalty-free stability results for DG methods do not extend directly because continuity prevents independent elementwise polynomial choices.The paper also notes that weakly imposed boundary conditions were used in LES-type turbulence computations, where the boundary-layer mean flow was more accurately captured than with strong imposition.Source: §1 · p. 1
  • A penalty-free method removes the penalty parameter and avoids Lagrange multipliers, making it relevant to flow problems whose system matrices are already non-symmetric because of convection.The authors also cite interest in reducing free parameters without increasing coupling degrees of freedom.Source: §1 · p. 1
  • For the Poisson problem, the non-symmetric method is stable and optimally convergent in H1 for polynomial orders k ≥1, while its L2 error rate is suboptimal by half an order in h.The authors note this non-optimality is less important for continuous Galerkin methods than for DG methods.Source: §1 · p. 2
  • The paper applies its results to convection–diffusion equations, first through Streamline–diffusion and then by outlining an extension to the Continuous interior penalty method.Source: §1 · p. 2
  • The penalty-free formulation is not claimed to be the most accurate: the smallest L2 error is obtained with boundary penalty.The introduction also notes stabilizing properties for outflow layers, analyzed in and illustrated numerically in the paper.Source: §1 · p. 2
  • The proof establishes results for weak imposition of boundary conditions; extensions to domain decomposition and fictitious domains are described as straightforward, and some analysis is only sketched.The authors assume readers have basic familiarity with stabilized finite element analysis.Source: §1 · p. 2

2. The pure diffusion problem.

The paper uses the Poisson equation with prescribed source and boundary data as its model problem. Convexity is assumed when H2-regularity is needed.

2. The pure diffusion problem.

  • The model is the Poisson equation −∆u = f in Ω with u = g on ∂Ω.Source: §2 · p. 2
  • The domain is bounded in R2 with polygonal boundary, and convexity is assumed wherever H2-regularity of the exact solution is needed.Source: §2 · p. 2
  • The data satisfy f ∈ L2(Ω), while g is taken in H1/2(∂Ω) or H3/2(∂Ω); the weak formulation seeks u ∈ Vg.The continuous problem is well-posed by the Lax–Milgram lemma and has the stated H1 stability and, under the regularity assumptions, H2 bound.Source: §2 · pp. 2–3

3. The finite element formulation.

The finite element formulation uses continuous piecewise polynomials on fitted, quasi-uniform, shape-regular meshes. Its non-symmetric boundary terms permit the penalty parameter to be set to zero.

3. The finite element formulation.

  • The analysis uses fitted quasi-uniform, shape-regular triangulations, with at least five element faces on each boundary face Γi.The text says quasi-uniformity can be lifted at the cost of standard technicalities and reduced readability.Source: §3 · p. 3
  • The formulation uses continuous piecewise-polynomial finite element functions, with local polynomial degree at most k.Source: §3 · p. 3
  • The bilinear form adds −⟨∇u_h · n, v_h⟩∂Ω and +⟨u_h, ∇v_h · n⟩∂Ω boundary terms to a(u_h, v_h).Source: §3 · p. 3
  • The key formulation choice is to set the penalty parameter γ to zero without loss of stability or accuracy.Source: §3 · p. 3

4. Stability.

Without a penalty term, testing with the discrete solution does not immediately control its boundary trace. The stability proof recovers boundary control through an inf-sup argument.

4. Stability.

  • The penalty term’s role is to provide enough boundary-trace control for the Poincaré inequality and well-posedness; without it, testing with u_h does not immediately control the boundary term.Source: §4 · p. 4
  • The proof treats the normal gradient ∇u_h · n as a Lagrange multiplier and constructs test functions to control solution averages on boundary patches.The patches group elements touching the boundary by a face or vertex.Source: §4 · p. 4
  • A technical lemma supplies the inf-sup condition for the penalty-free formulation.The patch construction assumes each boundary portion Fj has at least four inner nodes, with the text identifying a limited corner-element exception.Source: §4 · pp. 5–6

5. A priori error estimates.

The section derives a priori error estimates using stability, Galerkin orthogonality, continuity, and interpolation. For the L2 estimate, it introduces an adjoint problem and uses regularity of its solution.

5. A priori error estimates.

  • The stability estimate and Galerkin orthogonality lead to error estimates in the ||·||1,h norm.Continuity of the bilinear form is established as part of the argument.Source: §5 · p. 7
  • The approximation argument uses the Scott-Zhang interpolant, the triangle inequality, consistency, and continuity.The cited passages describe the proof steps but do not provide the resulting inequality.Source: §5 · p. 8
  • Lack of adjoint consistency causes suboptimal L2 convergence, though using the method on the domain scale may reduce the suboptimality to h.The passage contrasts the domain scale with the element scale used in DG methods.Source: §5 · p. 8
  • The L2-error argument uses an adjoint problem with right-hand side u − uh and homogeneous Dirichlet data.The proof then invokes H2 regularity of the adjoint solution.Source: §5 · pp. 8–9

6. The convection–diffusion problem.

This section extends the penalty-free non-symmetric Nitsche analysis to convection–diffusion, treating diffusion- and convection-dominated flows and reporting stability and accuracy results. It also establishes local stability and continuity control for the Continuous Interior Penalty construction, noting additional orthogonality arguments needed to extend the interpolant construction to higher-order elements.

6. The convection–diffusion problem.

  • The convection–diffusion analysis assumes nonzero normal flow on a boundary portion of positive measure, and adds stabilization in the convection-dominated regime.The method is motivated by flow problems, whose advection term already makes the system non-symmetric; the normal flow need not be nonzero across the entire boundary.Source: §6 · pp. 9–10

6.1. Diffusion dominated regime Pe < 1.

  • For diffusion-dominated flow (Pe < 1), the method has an inf-sup condition and optimal a priori estimates in the 1,h norm, assuming σ = 0.The analysis uses a strengthened norm that includes diffusion and boundary-flow contributions.Source: §6.1 · pp. 10–11, §6 · p. 10

6.2. Convection dominated regime: the Streamline–diffusion mehod.

  • For convection-dominated flow (Pe > 1), streamline-diffusion stabilization yields optimal convergence under interpolation hypotheses (H1) and (H2).The proposed stability norms are insufficient for general optimal error estimates, so the proof instead uses a specially constructed interpolant; the stabilization parameter is δ = γSDh/|β|.Source: §6.2 · pp. 12–13

6.2.1. Convection dominated regime: the Continuous Interior Penalty

  • For continuous interior penalty stabilization with cσ > 0, the analysis gives quasi-optimal L2 convergence when a suitable interpolant exists, and establishes that interpolant for piecewise affine elements.The interpolant must satisfy the approximation, normal-gradient, and additional design conditions.Source: §6.2.1 · pp. 15–16
  • The interpolant πCIP u is constructed by choosing ri = ∇u · n.Source: §6.2.1 · p. 16
  • The local stability estimate follows from invertibility of a 2×2 matrix and scaling back to the physical patch, with the patch norm controlled by the data norm.The matrix is invertible because its first-row coefficients are positive, while the second-row coefficients have opposite signs.Source: §6.2.1 · p. 17
  • Continuity control uses a quasi-interpolant based on node averages of β · ∇vh, together with bounds involving the CIP jump term and elementwise constant projection.The cited bounds compare β · ∇vh with its quasi-interpolant and its projection onto piecewise constants on each patch.Source: §6.2.1 · p. 17
  • On each element, β · ∇vh is constant; the passage gives this as the reason the second claim holds, while saying the first claim was proved in.Source: §6.2.1 · p. 18

7. Numerical examples.

The numerical examples test smooth-solution accuracy and convection-diffusion behavior. They report comparable H1 errors to strongly imposed conditions, little H1 sensitivity to the penalty parameter, and a stabilizing effect from weak boundary imposition in convection-dominated flow.

7.1. Problem with smooth solution.

  • On the smooth-solution example, symmetric Nitsche produces boundary oscillations on coarse meshes that vanish on finer meshes.The non-symmetric method has optimal convergence in this example, though the symmetric method’s stability is unlikely on general meshes.Source: §7.1 · p. 18, Figure 7.1 · p. 20
  • For piecewise quadratic approximation, the non-symmetric method’s L2 error is twice that of strongly imposed conditions, while their H1 errors are comparable.The comparison is reported for unstructured meshes; the cited table caption identifies the methods and approximation.Source: §7.1 · p. 19, Table 7.2 · p. 19
  • Across penalty parameters γ = 0, 10, 20, 40, 80, the L2 error changes visibly but negligibly, while the H1 error is unaffected.The study reports results for piecewise affine and quadratic approximations.Source: §7.1 · p. 19, Table 7.3 · p. 19, Table 7.4 · p. 20

7.2. Problem with outflow layer.

  • The weakly imposed boundary condition strongly stabilizes the convection-diffusion solution without added stabilization; SD and CIP stabilization remove remaining spurious oscillations.The comparisons use N = 80 piecewise affine approximations for the unstabilized cases and N = 80 piecewise quadratic approximations for the stabilization comparison.Source: §7.2 · p. 19, Figure 7.2 · p. 20, Figure 7.3 · p. 20

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