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Some Error Analysis on Virtual Element Methods
Long Chen, Jianguo Huang
TL;DR
The paper addresses the need for detailed error analysis of virtual element methods on polygonal meshes. It uses finite-element analysis and compactness under a virtual quasi-uniform triangulation assumption, establishing inverse inequalities, norm equivalence, and interpolation estimates while excluding high-aspect-ratio polygons.
Problem
Detailed justifications for inverse estimates, norm equivalence, and interpolation error estimates in VEM analysis were not presented in earlier work.
Method
The paper analyzes VEM spaces using finite-element mathematical tools, compactness-based scaling, and polygonal meshes admitting uniformly shape-regular, quasi-uniform virtual triangulations.
Results
The paper establishes inverse inequalities, norm equivalence between VEM function norms and degrees-of-freedom representations, and interpolation error estimates for several VEM spaces.
Takeaways & Limitations
The results provide error-analysis foundations for VEM spaces on polygonal meshes satisfying the virtual triangulation assumption.
Takeaways & Limitations
The assumption excludes high-aspect-ratio polygons because the resulting constants are not robust to element aspect ratio.
Abstract
from arXiv · showhide
Some error analysis on virtual element methods including inverse inequalities, norm equivalence, and interpolation error estimates are presented for polygonal meshes which admits a virtual quasi-uniform triangulation.
1. Introduction
The paper develops inverse inequalities, norm equivalence, and interpolation estimates for VEM spaces on polygonal meshes admitting uniformly shape-regular, quasi-uniform virtual triangulations. It replaces generalized scaling arguments with finite-element analysis tools and compactness-based reasoning under explicit mesh assumptions.
- Motivation: Virtual element analysis requires inverse inequalities, norm equivalence, and approximation estimates, but earlier results lacked detailed justifications.These estimates are fundamental in finite element theory and likewise support theoretical analysis of VEMs.
- Method: The compactness argument rescales polygons to unit diameter, treats admissible shapes as a compact set, and obtains uniform constants when the relevant functional is continuous.The continuity step is used to show that f(K) attains a maximum.
- Method: For virtual element spaces, continuity of Poisson solutions with respect to polygon shape is required, making the generalized scaling route technically subtle.The paper instead seeks finite-element-style proofs under the virtual-triangulation assumptions.
- Mesh assumptions: The paper assumes each polygon is simple and admits a virtual triangulation that is uniformly shape regular and quasi-uniform, with mesh size proportional to the polygon diameter.Each polygon edge must be a side of a triangle in the auxiliary triangulation.
- Method: Polygonal scaling is handled by decomposing each polygon into shape-regular triangles rather than mapping general polygons to a nonexistent reference polygon.This allows estimates to be transferred from triangle-level arguments to polygonal elements.
- Results: The resulting analysis establishes inverse inequalities, norm equivalence with degrees of freedom, and interpolation error estimates for several VEM spaces.The introduction states norm equivalence in terms of hK∥χ(v)∥l2 and ∥v∥0,K, and introduces canonical-interpolant estimates.
2. Virtual Element Methods
The paper formulates virtual element spaces on polygonal meshes, using degrees of freedom and polynomial projections rather than explicit interior point values. Its framework relies on virtual triangulations and establishes the computational and stability setting for later analysis.
- The analysis considers a two-dimensional domain decomposed into a polygon mesh satisfying assumption A1.
- Assumption A2 supplies each polygon with a shape-regular, quasi-uniform virtual triangulation whose triangle sizes are comparable to the polygon size.The number of triangles is uniformly bounded, and constants depend on the triangulation's shape regularity and quasi-uniformity.
- VEM shape functions may be implicitly defined through an interior PDE, while their degrees of freedom suffice to assemble an accurate and stable numerical method.This avoids requiring pointwise values and gradients inside each polygon for stiffness-matrix assembly.
- The degrees of freedom include boundary traces or edge moments and element moments, with polynomial-degree ranges determined by k and l.The framework also introduces scaled monomials and a dual space of functionals for representing these degrees of freedom.
- A VEM function can be identified with its degree-of-freedom vector, while distinct spaces may share the same degrees of freedom but represent different functions.Unisolvence is justified by matching dimensions and proving uniqueness from vanishing degrees of freedom.
- For stability, the gradient projection must be supplemented by a constraint because the gradient inner product has a constant kernel.Using the projection alone does not produce a stable method; the paper later connects stability to inverse and Poincaré–Friedrichs inequalities.
3. Inverse Inequalities
The paper derives inverse inequalities for VEM functions on polygons by combining virtual triangulations, projection stability, and an H1-orthogonal decomposition. The resulting constants depend only on the auxiliary triangulation's regularity and quasi-uniformity.
- A simple scaling argument does not clarify how norm-equivalence constants depend on polygon shape, motivating a virtual-triangulation analysis.
- The virtual triangulation enables use of finite-element estimates on triangles while retaining the VEM condition that the Laplacian belongs to a polynomial space.
- The polynomial inverse inequality on a polygon is obtained by restricting the polynomial to each virtual-triangulation triangle and scaling locally.The resulting constant depends only on the shape regularity and quasi-uniformity of the virtual triangulation.
- The operator QK preserves boundary values and matches interior moments against the finite-element subspace with zero boundary nodes.Its weighted stability is established by decomposing QKv into boundary and interior components and estimating the boundary contribution with weighted trace bounds.
- Every H1 function is decomposed into a harmonic trace part and a zero-boundary nonzero-moments part, and the decomposition is H1-orthogonal.The energy identity expresses the squared gradient norm as the sum of the two component gradient norms.
- Theorem 3.6 establishes an inverse inequality for VEM functions with a constant depending only on the virtual triangulation's shape regularity and quasi-uniformity.The proof combines the weighted harmonic-part estimate with the inverse estimate for the nonzero-moments part.
- The inverse inequality yields L2-stability for the projection QK and for the gradient projection restricted to VEM spaces.
4. Norm Equivalence
The paper establishes norm equivalences for VEM functions and their degree-of-freedom vectors on polygons using virtual quasi-uniform triangulations. It extends these results to stabilization formulations and spaces requiring no additional moment degrees of freedom.
- 4.1–4.2. Norm equivalence for VEM spaces: The analysis proves L2–l2 norm equivalence between a VEM function and its degree-of-freedom vector.The result is developed through polynomial norm equivalence, boundary and interior degree-of-freedom estimates, harmonic decomposition, and Poincaré inequalities.
- 4.1. Polynomial norm equivalence: Polynomial norm equivalence is obtained without a reference polygon by scaling a uniformly interior circle from the auxiliary triangulation.The resulting constant depends only on the shape regularity and quasi-uniformity constants of the virtual triangulation.
- 4.2. Norm equivalence for VEM spaces: Theorem 4.5 establishes norm equivalence for functions in Vk,l(K), while Corollary 4.6 extends it to Wk(K) without additional moment degrees of freedom.For Wk(K), the additional moments needed in the larger space are bounded through the existing degrees of freedom.
- 4.3. Norm equivalence of VEM formulation: The formulation-level norm equivalence supports stability of two VEM stabilization choices.For Wk(K), moment degrees of freedom vanish after the relevant projection, so the stabilization can use boundary degrees of freedom only.
5. Interpolation Error Estimates
The paper constructs several interpolants for VEM spaces and proves optimal-order interpolation error estimates in both L2 and H1 norms. The proofs compare VEM interpolants with polynomial and auxiliary finite-element interpolants while exploiting shared degrees of freedom and projection properties.
- 5.1. Projections and interpolants: The interpolation framework introduces an L2 polynomial projection, an auxiliary finite-element nodal interpolant, and VEM interpolants defined through local problems or canonical degrees of freedom.These objects provide the comparison chain used for the subsequent VEM error estimates.
- 5.2. Interpolation estimate of uI: The interpolant uI satisfies an optimal-order error estimate in both L2 and H1 norms.The proof compares uI with the auxiliary finite-element interpolant and uses Poincaré and energy identities.
- 5.3. Interpolation estimate of IKv: The canonical interpolant IKv has an optimal-order error estimate in both L2 and H1 norms for v ∈ Hk+1(K).Moment preservation makes the Laplacian term orthogonal, while inverse estimates control the remaining polynomial contribution.
- 5.4. Interpolation estimate of IWk v: The interpolant IWk v also achieves an optimal-order error estimate in both L2 and H1 norms for v ∈ Hk+1(K).The proof bridges IWk v and IKv through their shared degrees of freedom and projection relations, although they belong to different spaces.
6. Conclusion and Future Work
The paper establishes inverse inequalities, norm equivalences, and interpolation error estimates for several VEM spaces on polygons admitting virtual quasi-uniform triangulations. Its scope excludes high-aspect-ratio polygons, while three-dimensional extension remains future work.
- Conclusion: The paper establishes inverse inequalities, norm equivalence with degree-of-freedom vectors, and interpolation error estimates for several VEM spaces under assumption A2.Assumption A2 requires a virtual quasi-uniform triangulation of the polygon.
- Scope and limitations: Assumption A2 rules out polygons with high aspect ratio, and the analysis constants are not robust to the element aspect ratio.For rectangles, the number of shape-regular subelements in the auxiliary decomposition depends on hmax/hmin.
- Future work: The proofs are presented in two dimensions; extension to three-dimensional polyhedra is identified as ongoing work.The proposed route applies A2 to each face and then to the polyhedron itself, but the details remain undeveloped.