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Virtual Element Methods on Meshes with Small Edges or Faces
Susanne C. Brenner, Li-yeng Sung
TL;DR
Virtual element methods are observed numerically to retain convergence on polygonal or polyhedral meshes with small edges or faces, but error estimates were needed to justify this behavior. The paper develops such estimates under general shape-regularity assumptions and establishes optimal convergence under stated additional conditions.
Problem
The paper addresses the need for error estimates justifying observed virtual element convergence on meshes containing small edges or faces.
Method
The paper develops virtual element error estimates under star-shapedness, uniformly bounded subdomain edges or faces, and suitable face regularity assumptions.
Results
Optimal convergence is established in two dimensions under shape regularity for one stabilizing form, and in three dimensions when edges of each face are mutually comparable.
Takeaways & Limitations
The estimates justify existing numerical results for virtual element methods on polygonal or polyhedral meshes with small edges or faces.
Takeaways & Limitations
The three-dimensional analysis assumes star-shaped polyhedra, uniformly bounded faces, and faces satisfying the stated two-dimensional shape regularity conditions.
Abstract
from arXiv · showhide
We consider a model Poisson problem in $\R^d$ ($d=2,3$) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges ($d=2$) or small faces ($d=3$).
1. Introduction
The paper studies virtual element methods for Poisson problems on polygonal or polyhedral meshes containing small edges or faces. It develops error estimates under general shape-regularity assumptions to justify observed convergence behavior.
- Scope: The paper treats two- and three-dimensional Poisson problems and concludes with remarks on the resulting error estimates.The analysis is organized around a star-shaped condition, followed by separate two- and three-dimensional treatments.
- Motivation: Numerical experiments indicate that virtual element convergence rates do not noticeably deteriorate when meshes contain small edges or faces.The paper aims to establish analytical estimates supporting these observations.
- Shape regularity assumptions: The analysis assumes star-shaped elements with uniformly bounded numbers of edges or faces, plus two-dimensional shape regularity for three-dimensional faces.These conditions apply separately to polygonal and polyhedral meshes.
- Main contribution: The resulting error estimates are optimal up to at most a logarithmic factor involving ratios over edges or edges of faces.The supplied passages identify the ratio structure but do not provide its complete displayed formula.
- Analytical setting: For technical simplicity, the domain is assumed convex so elliptic regularity gives the solution H2 regularity.This is an additional global assumption used in the analysis.
2. A Star-Shaped Condition
The section develops geometric and functional estimates from a uniform star-shaped condition. These estimates support trace, extension, polynomial, and Poincare-Friedrichs bounds in two and three dimensions.
- Star-shaped condition: Each element D is star-shaped with respect to a disc or ball whose radius is ρD hD, yielding BD ⊂ D ⊂ ˜BD.The concentric enlarged disc or ball ˜BD has radius hD.
- Geometric consequences: The star-shaped condition provides uniform geometric control, with hidden constants depending only on ρD, and additionally on k in later polynomial estimates.This dependence is stated explicitly for the section’s subsections.
- Approximation estimates: The geometry implies Bramble-Hilbert and Poincare-Friedrichs estimates used to control functions and polynomial approximations on elements.The section also derives trace inequalities on boundaries and faces.
- Lipschitz mapping: A Lipschitz isomorphism between the inscribed disc or ball and D has bounded map and inverse derivatives controlled only by ρD.This map underlies the geometric regularity and extension arguments.
- Trace inequalities: The section includes specialized trace inequalities for edges in two dimensions and faces in three dimensions.These estimates require separate derivations for boundary traces and are later used in the analysis.
- Trace and extension estimates: In two dimensions, a Calderon-Zygmund extension operator maps H1(D) and H2(D) functions into corresponding Sobolev spaces on R2 with controlled norms.Interpolation also yields an H3/2 extension estimate, which supports edge trace bounds.
- Polynomial estimates: Polynomial estimates are obtained through norm equivalence and scaling, while polynomial Laplacian estimates use a right inverse of the Laplacian.The right inverse maps Pk−2 to Pk and depends only on k.
3. Local Virtual Element Spaces in Two Dimensions
The paper defines local two-dimensional virtual element spaces through boundary polynomial traces, polynomial Laplacians, and projection constraints. It then establishes projection, minimum-energy, and maximum-principle tools for stability and error analysis.
- Boundary representation: Boundary traces use continuous piecewise polynomials on the edges, with edge length denoted he.The boundary space is built from restrictions to individual edges.
- Projection: The projection Πk,D maps H1(D) onto Pk(D) with respect to the section’s H1-based inner product.Its polynomial gradient is characterized through the corresponding projection identities.
- Virtual element space: The local space Qk(D) contains H1 functions whose boundary trace is piecewise polynomial of degree at most k and whose negative Laplacian lies in Pk(D).A third projection-related condition completes the definition.
- Degrees of freedom: The degrees of freedom consist of boundary nodal values determining the edge polynomials and interior moments of Π0,k−2,Dv.These degrees of freedom also permit computation of the relevant polynomial projections.
- Energy control: A minimum energy principle bounds the H1 seminorm of a virtual element function by that of a comparison function with matching boundary data and projection moments.This principle is used to control virtual element functions in the stability analysis.
- Auxiliary estimates: A maximum principle and polynomial estimates provide additional bounds used for interpolation, stability, and error analyses.The maximum-principle result is explicitly connected to later two- and three-dimensional analyses.
- Seminorm and stability: The section introduces a seminorm on H1(D) and derives its properties using bump functions, norm equivalence, scaling, and integration by parts.All hidden constants in the relevant subsection depend on ρD and k.
3.4. The Semi-norm ||| · |||k,D.
The seminorm subsection relates edge projections and polynomial estimates to stability bounds. Its constants are controlled by the element’s shape parameter and polynomial degree.
- Edge projection: The edge operator Πk−1,e is the L2(e)-orthogonal projection onto Pk−1(e).This projection enters the seminorm-related estimates.
- Estimate dependence: The estimates use standard one-variable polynomial bounds, with constants depending only on ρD and k.The stated dependence excludes mesh sizes from the hidden constants.
3.5. Estimates for Π
This section establishes stability estimates for the projection Π, including a bound involving the L2 norm and the seminorm |||·|||k,D.
- The estimates are derived by combining earlier relations, including (2.5), (3.3), (3.4), (3.12), and Lemma 3.7.
- The projection Π satisfies a stability estimate controlling ∥∇k,Dζ∥L2(D) by |||ζ|||k,D for ζ ∈ H1(D).
3.6. Estimates for Π0
This section develops stability estimates for Π0 and records that the hidden constants depend only on the domain parameter ρD and polynomial degree k.
- The subsection states stability estimates for Π0 whose hidden constants depend only on ρD and k.
- The estimates are obtained from relations (2.5), (3.18), and (3.19), together with preceding lemmas and inequalities.
- These estimates bound the H1 seminorm of a virtual element function v ∈ Qk(D) using projected quantities and boundary-data norms.
3.7. Inverse Estimates.
This section derives inverse estimates for virtual element functions, including bounds using boundary data and tangential derivatives, with constants controlled by mesh and degree parameters.
- Boundary-data norms are crucial for the stability analysis of virtual element methods in Section 4.2.
- Lemma 3.12 provides an estimate with a constant depending only on ρD and k.
- The inverse estimates use lifting operators, bump functions, and polynomial corrections to control virtual element functions.
- Tangential derivatives appear in the inverse estimates through ∂v/∂s along the boundary.
- Several constants additionally depend on |ED| and k, as stated in Lemmas 3.14 and Corollary 3.15.
3.8. The Interpolation Operator.
This section defines the interpolation operator Ik,D through shared degrees of freedom and derives stability and interpolation error estimates in Sobolev and L∞ norms.
- The section begins with stability estimates for the interpolation operator and extends them to three-dimensional face restrictions.
- Interpolation error estimates are derived for H2(D) and H3/2(D) functions, with constants depending on ρD, |ED|, and k.
- Lemma 3.20 gives the bound ∥Π1,DIk,Dζ∥L2(D) ≤ C hD^2|ζ|H2(D) for ζ ∈ H2(D).
- The section also establishes interpolation error estimates in the L∞ norm.
3.9. The Null Space of Π
This section develops inverse estimates for polynomial components associated with the null space of Π, using polynomial right inverses of the Laplacian and boundary inequalities.
- Null-space estimates: The resulting estimates yield inverse bounds for functions in the relevant null space of Π.The section states that estimates (3.30) and (3.31) can be simplified using the preceding lemmas.
- Boundary control: For boundary polynomials vanishing at one point, tangential derivatives control the function through a Poincaré–Friedrichs inequality on ∂D.The constant in this boundary estimate depends only on the polynomial degree k.
- Constant dependence: Some estimates have hidden constants that additionally depend on the number of edges |ED|.This dependence is explicitly distinguished from estimates whose constants depend only on ρD and k.
4. The Poisson Problem in Two Dimensions
The two-dimensional analysis defines virtual element discretizations for the Poisson problem on polygonal meshes and derives energy, L2, and edgewise L∞ error estimates under stated mesh assumptions.
- Problem and discretization: The method uses polygonal triangulations of a convex polygon, with a global virtual element space assembled from local spaces and a mesh-dependent piecewise H1 framework.The mesh parameter is h = maxD∈Th hD, and discontinuous piecewise polynomials provide the comparison space.
- Mesh assumptions: The analysis assumes uniform lower bounds on element star-shapedness and a uniform upper bound N on the number of edges per element.Under these assumptions, hidden constants depend only on ρ, N, and k.
- Discrete formulation: The discrete problem uses a non-inherited symmetric positive definite bilinear form with one of two local stabilizing bilinear forms, yielding well-posedness through stability.The paper also relates the stabilizers to previously introduced forms and allows an alternative stabilizer.
- Abstract error estimate: The abstract energy-norm analysis controls consistency and approximation terms, including the load-related estimate involving Ξh.The choice of Ξh is linked to the estimate for the load term.
- Energy-norm estimates: For u ∈ Hℓ+1(Ω), 1 ≤ ℓ ≤ k, Theorems 4.5 and 4.6 provide concrete energy-norm estimates for uh and computable polynomial approximations, with constants depending on ρ, N, and k.The estimates involve αh defined in (4.11).
- L2 estimates: The paper derives corresponding L2 estimates for uh and computable approximations, together with consistency estimates under the same regularity range.These results again use αh and constants depending on ρ, N, and k.
- Edgewise L∞ estimates: Edgewise L∞ estimates are established for computable approximations; one stabilizer requires quasi-uniformity, while the other yields a result without that assumption.The quasi-uniform case introduces γ through hD ≥ γh.
- Comparison of stabilizers: Theorem 4.15 gives an L∞ estimate for the computable approximation associated with the second stabilizer without quasi-uniformity, whereas the analogous first-stabilizer result assumes quasi-uniformity.The corresponding quasi-uniform theorem has constants depending on ρ, N, γ, and k.
5. Virtual Element Methods for the Poisson Problem in Three Dimensions
The three-dimensional analysis extends the two-dimensional virtual element framework to polyhedral meshes under shape-regularity assumptions, including computable projections, interpolation, stability, and error estimates. The resulting constants depend on mesh regularity, polynomial order, and βh, so small faces themselves do not affect performance, while relative edge sizes on faces remain relevant.
- Framework: The three-dimensional analysis follows the two-dimensional strategy, with details supplied only for estimates requiring different derivations.Many results from Sections 3 and 4 carry over by identical arguments.
- Framework: Polyhedral meshes require star-shaped elements, uniformly bounded numbers of faces, and shape-regular faces.The assumptions use parameters ρ and N independently of the mesh size.
- Virtual element space: The local virtual element space consists of H1 functions with continuous piecewise-polynomial traces, polynomial Laplacian, and an additional orthogonality condition.Its degrees of freedom include edge values and face and element moments, from which the relevant projections are computable.
- Interpolation and stability: The interpolation operator matches the degrees of freedom and reproduces every polynomial in Pk(D).Its error estimates are obtained from Bramble–Hilbert bounds and stability estimates under the stated regularity assumptions.
- Error estimates: The discrete problem is well-posed, and the method admits analogous energy- and L2-norm error estimates for exact and computable approximations.The corresponding theorems assume u ∈ Hℓ+1(Ω), with 1 ≤ ℓ ≤ k, and constants controlled by ρ, N, and k together with βh.
- Error estimates: Small faces do not affect method performance; only the relative sizes of edges on each face enter the error-estimate constants through βh.The L∞ estimates additionally require a quasi-uniform mesh and introduce γ into the constant dependence.
6. Concluding Remarks
The paper establishes error estimates justifying optimal convergence for virtual element methods on meshes with small edges or faces, under stated shape-regularity conditions. It also identifies extensions to other stabilizations, higher-order methods, and general polygonal or polyhedral domains.
- Error estimates justify existing numerical evidence for virtual element methods on polygonal or polyhedral meshes with small edges or faces.
- In two dimensions, the method using SD2 is optimal under the Section 4.1 shape-regularity assumptions.
- In two dimensions, the method using SD1 is optimal when edges within each polygonal subdomain are comparable.
- In three dimensions, optimal convergence additionally requires comparable edges on every face of the polyhedral mesh.
- The results extend to alternative stabilizing bilinear forms, for which stability is automatic and error analysis introduces no new difficulties.
- The techniques also extend to higher-order virtual element methods and yield error estimates on general polygonal or polyhedral domains.