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The fully nonconforming virtual element method for biharmonic problems
P. F. Antonietti, G. Manzini, M. Verani
TL;DR
The paper addresses numerical approximation of biharmonic problems on polygonal meshes using a fully nonconforming virtual element method. The arbitrary-order method allows globally discontinuous approximation functions, achieves an optimal broken-energy error estimate, and is numerically consistent with the theoretical convergence rates.
Problem
The paper addresses numerical approximation of fourth-order elliptic biharmonic problems on polygonal meshes, where the method must accommodate polygonal elements and nonconforming approximation.
Method
The authors introduce an arbitrary-order fully nonconforming virtual element discretization whose approximation space may contain globally discontinuous functions.
Results
The method has an optimal error estimate in a broken energy norm, and numerical results assess the validity of the theoretical estimate.
Takeaways & Limitations
The approach provides arbitrary-order accurate biharmonic approximation on polygonal meshes without requiring global C0 regularity.
Takeaways & Limitations
The analysis requires mesh regularity assumptions, and the algebraic problem becomes increasingly ill-conditioned.
Abstract
from arXiv · showhide
In this paper we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic problems. The approximation space is made of possibly discontinuous functions, thus giving rise to the fully nonconforming virtual element method. We derive optimal error estimates in a suitable (broken) energy norm and present numerical results to assess the validity of the theoretical estimates.
1. Introduction
The paper develops a fully nonconforming virtual element method for biharmonic problems on polygonal meshes, extending arbitrary-order approximation without global C0 continuity. It derives optimal broken-energy error estimates and validates them numerically.
- The proposed method approximates biharmonic problems on unstructured polygonal meshes with arbitrary order and no global C0 regularity.
- The fully nonconforming virtual element space may contain globally discontinuous functions, distinguishing the method from globally C0-nonconforming approaches.
- On triangular meshes, the lowest-order method reduces to the Morley element, while higher orders produce a new family of discontinuous nonconforming finite elements.
- The paper derives an optimal error estimate in a broken energy norm and uses numerical experiments to assess the theoretical estimate.
2. The continuous problem
The section formulates the clamped thin-plate problem on a convex polygonal domain and establishes the variational setting, including uniqueness and boundary operators used later.
- The plate occupies a convex polygonal domain, carries a transversal load, and has clamped boundary conditions under the Kirchhoff–Love model.
- The bending rigidity is D = Et3 12(1−ν2), determined by thickness, Young’s modulus, and Poisson’s ratio.
- The continuous problem is expressed variationally by seeking u ∈V, with duality pairing between V and its dual V ∗.
- Boundary conditions and the Poincaré inequality make the specified V-norm a norm, supporting well-posedness of the formulation.
- The variational problem has a unique solution u ∈V.
- The section defines bending-stress quantities Mnn, Mnt, and T, then uses integration by parts to derive identities involving polygonal edge boundaries and endpoint orientations.
3. Nonconforming virtual element discretization
The paper constructs an arbitrary-order fully nonconforming virtual element discretization on polygonal meshes, using locally defined spaces whose global functions may be discontinuous. The method supplies computable, stable bilinear forms through polynomial projections and degrees of freedom, and establishes the broken energy seminorm framework needed for analysis.
- 3.1. Technicalities.: The discretization uses an arbitrary-order nonconforming space Vh,ℓ and approximates the continuous bilinear form and load term with ah and ⟨fh, ·⟩.The method is posed on polygonal meshes satisfying regularity assumptions needed for convergence and error estimates.
- 3.1. Technicalities.: The broken H2-seminorm is a norm on the nonconforming space, providing the energy-norm structure used in the subsequent error analysis.The norm property follows from a broken H1-to-H2 estimate together with the stated nonconforming-space framework.
- 3.2. Local and global nonconforming virtual element space.: The local degrees of freedom are unisolvent, so vanishing degrees of freedom imply that the corresponding virtual function is zero.The proof uses the polynomial structure of the biharmonic operator and boundary terms, then removes the linear-polynomial kernel through vertex values.
- 3.2. Local and global nonconforming virtual element space.: The global virtual element space permits piecewise discontinuous functions and does not require global C0 continuity across the mesh.This fully nonconforming construction is assembled from local spaces while retaining continuity at internal vertices and prescribed edge conditions.
- 3.3. Construction of the bilinear form.: The elliptic projection onto Pℓ(K) is computable from the degrees of freedom, enabling evaluation of the local bilinear form and load approximation.Computability follows by expressing the terms in the integration-by-parts formula through polynomial volume and edge quantities represented by the degrees of freedom.
- 3.3. Construction of the bilinear form.: The local bilinear form is designed to satisfy polynomial consistency and stability, with a scaled degree-of-freedom inner product providing a practical stabilization choice.The stabilization is symmetric and positive definite, and the Euclidean scalar product of scaled degrees of freedom is identified as a simple implementation.
4. Error estimates
The analysis establishes existence, uniqueness, interpolation properties, and an optimal broken-energy-norm error estimate under mesh regularity and solution smoothness assumptions.
- The mesh assumptions require star-shaped elements, edges with controlled minimum length, and a shape-regular sub-triangulation from an interior point.The additional sub-triangulation condition is specifically required for the error analysis.
- The interpolant is defined by matching all global degrees of freedom and satisfies optimal approximation estimates for derivative orders s = 0, 1, 2.The estimate applies for 3 ≤ β ≤ k + 1 and scales as h^(β−s)|w|_β,Ω.
- Under mesh regularity assumptions, the discrete problem has a unique solution in the virtual element space.
- The proof uses coercivity and continuity of the discrete bilinear form together with interpolation and trace estimates to control the discrete error.For higher orders, the conformity analysis exploits edge moment conditions and integration by parts.
- The error analysis combines interpolation, consistency, and non-conformity terms to obtain the final broken-energy-norm estimate.The proof bounds each contribution after decomposing the conformity error and using standard interpolation estimates.
5. Numerical results
Numerical experiments on several unstructured polygonal mesh sequences test convergence, consistency, and the predicted dependence on polynomial degree.
- The study solves a clamped biharmonic problem on the unit square using four sequences of unstructured meshes and compares errors against mesh size and degrees of freedom.The mesh families include perturbed square, triangular, and non-convex polygonal constructions.
- For order ℓ = 2 on criss-cross triangular meshes, the nonconforming virtual element method coincides with the Morley finite element method.Criss-cross meshes split square cells into four triangular subcells along the diagonals.
- For ℓ = 5, computations stop after the fifth mesh because increasing algebraic ill-conditioning causes rounding errors to affect accuracy.
- The experimental convergence rates agree perfectly with the theoretical rates for all reported polynomial degrees and calculations.For Error2,h, slopes are expected near ℓ−1 versus h and (ℓ−1)/2 versus degrees of freedom.
- Polynomial consistency tests for orders 2 through 5 produce errors comparable to arithmetic precision, confirming exactness for polynomial solutions up to degree ℓ.The tests use monomial forcing terms over a wider collection of polygonal meshes.
6. Conclusions
The paper presents an arbitrary-order fully nonconforming virtual element method for biharmonic problems on polygonal meshes and derives an optimal broken-energy-norm estimate validated numerically.
- The method achieves arbitrary-order accuracy for biharmonic problems on polygonal meshes using possibly globally discontinuous virtual element functions.
- The numerical study uses four mesh families, including criss-cross triangular, mainly hexagonal, non-convex regular, and randomized quadrilateral meshes.
- Numerical results assess the validity of the optimal error estimate in the broken energy norm.
Appendix
The appendix documents mesh geometry and virtual element degrees of freedom across four mesh sequences, with tabulated refinement data and author affiliations.
- Tables 1–4 report refinement levels, polygonal cells, faces, vertices, mesh size h, and associated VEM degrees of freedom.
- The appendix includes geometric-data tables for criss-cross, remapped hexagonal, nonconvex octagonal, and randomized quadrilateral mesh sequences.
- The appendix lists G. Manzini’s affiliations with Los Alamos National Laboratory and the CNR institute in Pavia.