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Crouzeix--Raviart--Marini realisation of computable a priori $L^2$-error bounds on anisotropic meshes

Hiroki Ishizaka

arXiv:2609.06057v1math.NAcs.NA

TL;DR

Conforming Aubin–Nitsche estimates rely on elliptic regularity, which is inconvenient on non-convex domains where L2 data need not yield an H2 solution. The paper combines the Marini relation and hypercircle estimates for the lowest-order CR–RT pair to obtain computable L2 bounds. On the considered algebraically graded L-shaped family, q > 3/2 yields κM,h = O(h) and computable O(h^2) L2-operator bounds.

  • Problem

    Conforming Aubin–Nitsche estimates rely on elliptic regularity, which is inconvenient on non-convex polygons and polyhedra where L2 data need not produce an H2 solution.

    Source: §1 · p. 1

  • Method

    The paper combines the Marini relation and regularity-free hypercircle estimates for the lowest-order CR–RT pair, characterizing the squared Marini constant through a scalar generalized eigenvalue problem.

    Source: §1 · pp. 1–2

  • Results

    q > 3/2 gives κM,h = O(h) and computable O(h^2) L2-operator bounds on the considered algebraically graded L-shaped family.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The defect representation separates the CR–conforming energy gap from a geometric term and identifies which part controls loss of rate on the studied mesh families.

    Source: §1 · p. 1

    The rate theorem concerns the specified corner and grading mechanism, not arbitrary non-convex domains or graded meshes.

    Source: §2 · p. 28

Abstract

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We consider conforming $\Pone$ and lowest-order Crouzeix--Raviart (CR) approximations of the Dirichlet Poisson problem in two and three dimensions. We obtain computable a priori $L^2$-error bounds on anisotropic simplicial meshes without a global $H^2$-regularity assumption. For piecewise constant data, the Marini relation gives an equilibrated lowest-order Raviart--Thomas (RT) flux from the scalar CR solution. The square of the resulting Marini constant is the largest generalised eigenvalue of a problem involving only scalar conforming and CR matrices. Combined with the elementwise Poincar\'e constant, it controls both the conforming and CR errors. The CR $L^2$ estimate uses positivity of the discrete CR--conforming gap operator rather than a nonconforming Aubin--Nitsche argument. We also derive an exact decomposition of the Marini defect into the CR--conforming energy gap and an explicit geometric term with no aspect-ratio factor. On the algebraically graded L-shaped mesh family considered here, $q>3/2$ yields $\kappa_{M,h}=O(h)$ and computable $O(h^2)$ $L^2$-operator bounds.

1. Introduction

The paper develops computable L2 error bounds for conforming P1 and lowest-order CR approximations without requiring global H2 regularity. Its CR–Marini combination yields a scalar characterization of the relevant constant, an exact defect decomposition, and mesh-family results.

1. Introduction

  • The approach addresses the reliance of conforming Aubin–Nitsche estimates on elliptic regularity, which can fail for L2 data on non-convex polygons and polyhedra.The paper treats the lowest-order CR–RT pair using the Marini relation.Source: §1 · p. 1
  • The contribution combines the regularity-free hypercircle estimate with the Marini relation for the lowest-order CR–RT pair.It decomposes the Marini defect into the CR–conforming energy gap and an explicit geometric term.Source: §1 · p. 1
  • The square of the Marini constant is characterized by a scalar generalized eigenvalue problem involving conforming and CR quantities.The formulation uses scalar conforming and CR matrices and a diagonal geometric term.Source: §1 · pp. 1–2
  • The same mesh-dependent quantity controls conforming and CR L2 errors, while the CR estimate uses positivity of the discrete gap operator.The proof does not use a nonconforming Aubin–Nitsche argument.Source: §1 · p. 2
  • For the algebraically graded L-shaped family, q > 3/2 gives κM,h = O(h) and computable O(h^2) L2-operator bounds.In three dimensions, the paper gives an anisotropic comparison under local W2,p assumptions, with p > 2 for the first-order H1 Lagrange estimate.Source: §1 · p. 2
  • Sharp rates on general graded tetrahedral meshes with corner and edge singularities remain open.The stated rates use mesh-family assumptions, not assumptions required for the single-mesh construction or defect identity.Source: §1 · pp. 2–3

2. Poisson problem and finite element setting

The setting is the Dirichlet Poisson problem on a bounded polyhedral Lipschitz domain, discretized on conforming simplicial meshes. The mesh assumptions do not impose angle or quasi-uniformity conditions, and elementwise Poincaré constants provide a local estimate.

2. Poisson problem and finite element setting

  • The problem is posed on a bounded polyhedral Lipschitz domain, with the L2 inner product used for scalar- or vector-valued functions.Source: §2 · p. 3
  • The conforming simplicial mesh is non-degenerate, with no minimum-angle, maximum-angle, or quasi-uniformity assumption imposed.Source: §2 · p. 3
  • The optimal mean-zero Poincaré constant is defined elementwise, and the Payne–Weinberger inequality supplies an upper bound on each simplex.The stated convexity is local to each simplex and imposes no convexity assumption on the domain.Source: §2 · p. 3

3. CR–Marini equilibrated fluxes

The Marini relation constructs an equilibrated lowest-order RT flux from a scalar CR solution for piecewise constant data. The resulting defect and hypercircle properties connect the reconstruction to computable error constants and their scalar finite-dimensional characterization.

3. CR–Marini equilibrated fluxes

  • The Marini flux belongs to H(div; Ω) and has the required elementwise divergence.Its normal component is continuous across interior faces.Source: §3 · p. 5
  • The exact Marini defect decomposition consists of the CR–conforming energy gap and an explicit geometric term.Both terms are defined on the physical mesh and involve neither an inradius nor a shape-regularity factor.Source: §3 · pp. 5–6
  • The Marini flux is the unique best equilibrated RT approximation in the stated problem for piecewise constant right-hand sides.Source: §3 · p. 7
  • The CR–Marini constant realizes the lowest-order hypercircle constant, with a corresponding simplicial quantity also identified in three dimensions.In two dimensions, the identified constant is the one used by Liu and Oishi and by Takayasu et al.Source: §3 · p. 8

4. Computable a priori energy and L2 bounds

The section develops mesh-dependent computable energy and L2 bounds for conforming P1 and CR approximations. The bounds avoid global H2 regularity, and the CR L2 estimate relies on positivity of the CR–conforming gap operator.

4. Computable a priori energy and L2 bounds

  • The local Poincaré constant and Marini constant combine into mesh-dependent bounds, with explicit diameter-based quantities available for numerical use.The explicit replacement uses h/π for the local Poincaré constant.Source: §4 · p. 8, §4.1 · pp. 9, 11

4.1. The conforming P1 approximation.

  • Conforming P1 L2 bounds require no convexity or global H2 regularity, and the Ritz error operator's L2 norm is the square of its energy-operator norm.The same mesh-dependent bound controls the energy-operator norms of the primal and dual Ritz errors.Source: §4.1 · pp. 9–11
  • For piecewise constant data, the primal oscillation term vanishes, while the dual estimate retains M_h because its right-hand side need not be piecewise constant.Thus, the primal CR–Marini estimate does not impose an additional restriction on the dual right-hand side.Source: §4.1 · p. 11

4.2. The Crouzeix–Raviart approximation.

  • The CR L2 estimate follows from positivity of the CR–conforming gap operator, not from a nonconforming Aubin–Nitsche argument.The bounds hold on any non-degenerate conforming simplicial mesh and require neither convexity nor global H2 regularity.Source: §4.2 · pp. 12–14

5. Finite-dimensional form of the Marini constant

The Marini constant is computed through a finite-dimensional generalized eigenvalue problem using scalar conforming and CR matrices plus explicit element geometry. Its representation includes the CR–conforming energy gap and a geometric contribution.

5. Finite-dimensional form of the Marini constant

  • The Marini constant is characterized by a finite-dimensional generalized eigenvalue problem that requires scalar conforming and CR matrices and explicit element geometry, but no assembled RT0 basis.The matrix-free alternative applies the gap operator through one conforming and one CR sparse solve, while the geometric matrix is diagonal.Source: §5 · pp. 15–16
  • The complete Marini defect matrix is the sum of the CR–conforming energy-gap matrix and a diagonal geometric matrix containing elementwise second moments.The gap matrix is positive semidefinite.Source: §5 · pp. 15–16
  • The reported floating-point computations do not provide certified upper enclosures for the Marini constant.Certification would require outward-rounded enclosures for matrix assemblies, linear solves, and the largest generalized eigenvalue.Source: §5 · p. 16

6. Anisotropic geometry of the computable constant

The section separates the Marini constant into an energy-gap contribution and an explicit geometric contribution, then compares its mesh-family behavior using directional interpolation. It gives a corner-grading result for the L-shaped domain while keeping the scope of those rates specific to the analyzed family.

6. Anisotropic geometry of the computable constant

  • The Marini defect’s geometric term records actual edge lengths without an inverse-altitude, minimum-angle, or aspect-ratio factor.The single-mesh estimates also require neither shape regularity nor a maximum-angle condition.Source: §6 · p. 17
  • The geometric coefficient is comparable to h, so slower decay of κ_M,h along a family must come from the CR–conforming energy gap.On uniformly semi-regular families, directional estimates are uniform in aspect ratio when the required regularity or graded-mesh estimates hold.Source: §6 · p. 19

6.1. Directional anisotropic comparison.

  • In two dimensions, directional interpolation comparison allows arbitrarily large aspect ratios when the semi-regular parameter remains controlled.The comparison depends on finite local interpolation quantities and has constants independent of mesh sizes and aspect ratios.Source: §6.1 · pp. 20–21
  • In three dimensions, the directional comparison requires local W^2,p regularity with p > 2 for the first-order H1 Lagrange estimate.The comparison does not require the global elliptic H2 estimate.Source: §6.1 · p. 21

2. Thus, in either case,

The section handles the L-shaped domain by separating its re-entrant-corner singularity and using algebraic corner grading. It identifies the grading threshold for the approximation argument and limits the rate result to this mesh family.

Recovery by algebraic corner grading without global H2-regularity.

  • The corner singular function is not in H2, so elements touching the re-entrant corner cannot use the usual H2-based interpolation estimate.Those elements are handled separately from the non-corner elements.Source: §2 · pp. 24–25
  • For q > 1/α the corner radial integral is uniformly bounded, while at q = 1/α it grows like log n.The L-shaped corner has α = 2/3.Source: §2 · pp. 27–28
  • The rate theorem concerns this corner and grading mechanism, not arbitrary non-convex domains or graded meshes.Sharp rates on general graded tetrahedral meshes with corner and edge singularities are left open.Source: §2 · p. 28

7. Numerical experiments

The experiments compare uniform, stretched, quasi-uniform, and graded meshes in two and three dimensions. They report near-second-order L2 rates in smooth and graded settings, with the CR–conforming gap dominating the Marini defect and the graded L-shaped family recovering first-order decay of κ_M,h.

7.1. Smooth square: uniform and strongly stretched meshes.

  • On uniform square meshes, γ_geo,h ≃ h and κ_M,h’s rate tends to one; both L2 rates approach two, the CR–conforming gap dominates, and the conforming majorant has lower effectivity.The spectral decomposition and error tables report these trends for the uniform family.Source: §7.1 · p. 30, Table 1 · p. 30, Table 2 · p. 30
  • On stretched square meshes, κ_M,h/h remains bounded as aspect ratio grows; the geometric term stays proportional to h, the gap dominates, and L2 rates approach two on uniformly semi-regular meshes.The reported rate claim applies to these semi-regular meshes, not arbitrary anisotropic families.Source: §7.1 · p. 31, Table 3 · p. 31, Table 4 · p. 31

7.2. L-shaped domain: quasi-uniform and graded meshes.

  • For quasi-uniform L-shaped meshes, κ_M,h’s rate moves toward α=2/3, with slower decay attributed to the CR–conforming gap; L2 rates are consistent with the corner-limited 4/3 rate.The geometric term remains first order, and no global H2-regularity constant is used.Source: §7.2 · p. 32, Table 5 · p. 32, Table 6 · p. 32
  • For q=2>3/2, κ_M,h=O(h) with observed rate near one despite AR_h rising from about 31 to 255; both L2 rates are close to two.The graded anisotropic L-shaped mesh results are consistent with the stated threshold for recovering the computable order.Source: §7.2 · pp. 32–33, Table 7 · p. 33, Table 8 · p. 33

7.3. Three-dimensional tetrahedral meshes.

  • In both tetrahedral families, γ_geo,h/h is almost constant and the CR–conforming gap is larger; CR rates approach two earlier than conforming P1 rates, with no 3D anisotropic rate theorem claimed.The conforming P1 rate approaches two on finer meshes after remaining pre-asymptotic on coarse levels.Source: §7.3 · p. 33, Table 9 · p. 34

8. Concluding remarks

The lowest-order CR–RT construction turns the Marini relation into a scalar finite element eigenvalue problem and an exact defect decomposition. It yields computable error bounds without global H2-regularity, while sharp rates on graded tetrahedral meshes remain open.

8. Concluding remarks

  • The CR–RT Marini relation reduces the equilibrated RT reconstruction to a scalar CR solve, and the defect splits into the CR–conforming energy gap plus an explicit geometric term.The square of κ_M,h is characterized by a scalar generalized eigenvalue problem.Source: §8 · p. 34
  • Together with the local Poincaré constant, the construction gives conforming P1 and CR error bounds without global H2-regularity; on the graded L-shaped family, q>3/2 yields κ_M,h=O(h) and computable O(h^2) L2-operator bounds.Source: §8 · p. 34
  • Sharp rates for graded tetrahedral meshes with corner and edge singularities remain open, and the reported numerical eigenvalue computations are not certified enclosures.Certified majorants would require verified enclosures.Source: §8 · p. 34

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