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The nonconforming virtual element method
B. Ayuso de Dios, K. Lipnikov, G. Manzini
TL;DR
High-order mimetic finite difference schemes on general polygonal and polyhedral meshes remain difficult to construct, especially in three dimensions. The paper develops a nonconforming VEM for second-order elliptic problems in two and three dimensions, analyzes its errors, and establishes algebraic equivalence with a family of mimetic finite difference schemes. The method supports arbitrary approximation orders and element shapes within the stated mesh framework.
Problem
Constructing high-order mimetic finite difference schemes for second-order elliptic problems on general meshes remains challenging, particularly in three dimensions.
Method
The paper develops a nonconforming virtual element method for second-order elliptic problems in two and three dimensions, formulated for general polygonal and polyhedral meshes.
Results
The method has optimal convergence estimates in the energy and L2 norms and is algebraically equivalent to mimetic finite difference methods from.
Takeaways & Limitations
The virtual element framework provides arbitrary-order schemes on shape-regular polygonal and polyhedral meshes, including non-convex and degenerate elements, while supporting theoretical analysis.
Takeaways & Limitations
The error estimate for the nonconformity term requires additional regularity, at least u ∈ H2(Ω), beyond what the model problem may possess.
Abstract
from arXiv · showhide
We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods.
1. Introduction
The paper develops a nonconforming VEM for second-order elliptic problems on general polygonal and polyhedral meshes, addressing construction challenges in existing methods. It provides a unified formulation, error analysis, and an algebraic connection to MFD schemes.
- Motivation: High-order MFD schemes remain challenging to construct in three dimensions without introducing many conformity-enforcing degrees of freedom.The difficulty motivates a more flexible virtual-element framework for general meshes.
- Motivation: VEM recasts MFD schemes as Galerkin formulations while retaining flexibility for general, including non-convex, polygonal and polyhedral elements.Its virtual local functions are not computed; implementation uses degrees of freedom and polynomial components.
- Contribution: The proposed nonconforming VEM is developed for second-order elliptic problems in two and three dimensions and contains the MFD schemes from.The construction is simultaneous across dimensions and approximation orders.
- Main contributions: The method gives a one-shot construction for any degree, spatial dimension, and element shape, avoiding special cases in nonconforming approximation.For selected triangular and quadrilateral cases, its degrees of freedom coincide with related nonconforming finite elements.
- Main contributions: The analysis proves optimal energy-norm estimates and, for k ≥2, optimal L2-norm estimates, while also providing theory for the associated MFD schemes.The paper focuses on the Poisson problem, with results extending to more general second-order problems with constant coefficients.
2. Continuous problem and basic setting
The paper formulates a Poisson problem on polygonal or polyhedral meshes and builds a nonconforming setting with controlled interelement continuity. Shape-regular partitions and a discrete Poincaré inequality provide the analytical framework for the broken formulation.
- Continuous problem: The model problem is posed on a bounded polygonal or polyhedral domain in two or three dimensions with f in L2(Ω).
- Continuous problem: The weak formulation seeks u in Vg satisfying a(u,v)=⟨f,v⟩ for every v in V, with a continuous and coercive bilinear form.Lax–Milgram then gives existence and uniqueness of the solution.
- Mesh setting: The mesh family partitions Ω into polygonal or polyhedral elements under geometric assumptions involving edge-to-element sizes and star-shapedness.The global mesh size h is the maximum element diameter.
- Nonconforming setting: The nonconforming space imposes moment conditions on jumps across edges or faces, using polynomial test functions of degree at most k−1.
- Nonconforming setting: Although discontinuous functions have only a broken H1 seminorm, a discrete Poincaré inequality makes it a norm on the nonconforming space.The inequality holds already for k=1 with a constant independent of h.
- Nonconforming setting: The k=1 nonconforming space has the minimal continuity required for the analysis, while the bilinear form is decomposed into component terms for the method.
3. Non-conforming virtual element method
The section constructs a nonconforming virtual element method on polygonal and polyhedral meshes, defining local and global spaces, degrees of freedom, and a computable bilinear form with consistency, stability, and approximation properties.
- Global formulation: The method is formulated as a Galerkin approximation using global nonconforming virtual spaces, a discrete bilinear form, and a discrete right-hand side.The construction proceeds from local and global spaces to the bilinear form and right-hand side.
- Local virtual element space: Local degrees of freedom are edge or face moments up to order k−1 and cell moments up to order k−2.This definition applies to polygonal and polyhedral elements for every integer k ≥ 1.
- Local virtual element space: The degrees of freedom are unisolvent for the local space, even though its functions are generally nonpolynomial.The proof uses the divergence theorem to show vanishing degrees of freedom imply zero gradient and then zero function.
- Global virtual element space: The same moment structure defines the global space, with shared edge or face degrees of freedom and element moments across the mesh.The global count matches the total number of global degrees of freedom, and local unisolvence implies global unisolvence.
- Discrete bilinear form and approximation: The discrete bilinear form is polynomially consistent and stable, with constants independent of mesh size but dependent on mesh shape regularity.The analysis also provides standard approximation estimates for the interpolant and L2-projection-based approximation errors.
4. Error Analysis
The analysis establishes well-posedness and energy- and L2-error estimates for the nonconforming VEM, while identifying consistency and nonconformity errors. The argument requires additional regularity for first-order nonconformity estimates, and a-posteriori estimation is left open.
- Error structure: The nonconforming space introduces a consistency error because its discrete test functions are not contained in the continuous space.The term Nh measures this failure and is analogous to the consistency component in the Strang lemma for nonconforming finite elements.
- Limitations: Obtaining at least a first-order estimate for Nh requires additional regularity, at least u ∈ H2(Ω), beyond what some model problems provide.The cited examples include f ∈ H−1(Ω), nonconvex domains, and problems with jumping coefficients.
- Limitations: A-posteriori error estimation is identified as feasible but remains outside the scope of the paper.The authors retain the existing proof strategy to present the method and its novelty more simply.
- Well-posedness: The discrete problem has a unique solution under the stated assumptions, with stability and continuity supplied by the discrete bilinear form.The proof applies coercivity, continuity, the Poincaré inequality, and the Lax–Milgram theorem.
- Energy-norm error: |u − uh|1,h ≤ Ch^min(k,s)(∥u∥1+s,Ω + ∥f∥s−1,Ω) gives the energy-norm error estimate.The estimate is obtained by combining interpolation, consistency, continuity, and nonconformity bounds.
- L2-norm error: The L2 error is bounded by h times energy and approximation errors, plus a data-approximation term involving h^2 + h^min(2,¯k+1).Here ¯k = max{k − 2, 0}, and the result is stated for convex domains with the theorem’s regularity assumptions.
5. Connection with the nonconforming MFD
The nonconforming VEM stiffness matrix separates consistency from stabilization, creating a family of schemes through the stabilization matrix. Because VEM and MFD share degrees of freedom and consistency conditions, their stiffness matrices differ only in stabilization and can represent one another.
- VEM matrix structure: The VEM stiffness matrix is the sum of a consistency term and a stabilization term.Auxiliary matrices B, D, G, Π∇, and eG are used to define the matrix representation.
- VEM matrix structure: Different admissible choices of stabilization matrix S generate a family of virtual element schemes.The choice of S is not unique, so schemes can differ while retaining the same consistency structure.
- VEM–MFD equivalence: VEM and MFD use the same degrees of freedom and therefore satisfy the same consistency and stability conditions.Their consistency matrices are uniquely determined by exactness on the same polynomial space.
- VEM–MFD equivalence: The VEM and MFD stiffness matrices can be mapped to one another by matching their stabilization matrices through S and U.Lemma 5.1 states both directions: every mimetic stabilization yields an S, and every VEM stabilization yields a U.
6. Conclusions
The nonconforming VEM supports arbitrary-order schemes on complex polygonal and polyhedral meshes while enabling optimal error estimates and algebraic equivalence with MFD methods.
- The method constructs arbitrary-order schemes for any degree k ≥1 on shape-regular polygonal and polyhedral meshes, including non-convex and degenerate elements.
- Unlike classical nonconforming finite elements, the VEM construction applies at once to every degree k ≥1 and element shape.
- The analysis establishes optimal convergence estimates in both the energy and L2 norms.
- The VEM is algebraically equivalent to the mimetic finite difference method from.