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Stability Analysis for the Virtual Element Method
Lourenco Beirao da Veiga, Carlo Lovadina, Alessandro Russo
TL;DR
The paper studies stability of the VEM bilinear form under more general mesh assumptions, including edges arbitrarily small relative to element diameters. It develops a weaker stability analysis for several stabilization forms and shows that boundary-only stabilization retains the relevant stability and convergence behavior, consistent with numerical tests.
Problem
The paper examines the stability properties of the VEM bilinear form sE(·, ·) under weaker edge assumptions that allow edges arbitrarily small relative to the element diameter.
Method
The paper develops a new convergence-proof strategy requiring weaker stability conditions, analyzes multiple stabilization forms, and uses sub-triangulations and extension results in its analysis.
Results
The analysis establishes stability results under more general mesh assumptions, obtains error bounds consistent with numerical tests, and finds that boundary stabilization removes small-edge oscillations for k = 1 but can be too strong.
Takeaways & Limitations
Boundary-only stabilization can be used in the VEM scheme while preserving stability and convergence properties, although its parameter may need reduction when it is too strong.
Takeaways & Limitations
For one stabilization choice, robust theoretical results require the stronger assumption A3, whose bound contains the factor hE/hm(E).
Abstract
from arXiv · showhide
We analyse the Virtual Element Methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter), can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the "classical" one introduced in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199-214], and a recent one presented in [Wriggers, P., Rust, W.T., and Reddy, B.D., A virtual element method for contact, submitted for publication]. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can be used in the VEM scheme without affecting the stability and convergence properties. The numerical tests are in accordance with the theoretical predictions.
1 Introduction
The paper develops a stability analysis for VEM bilinear forms under more general polygonal-mesh assumptions, including meshes with arbitrarily short edges. It also compares stabilization choices and shows that the internal stabilization contribution can be removed without harming stability.
- VEM extends finite element methods to general polygonal and polyhedral meshes.
- The paper targets stability analysis when edge lengths are not uniformly comparable to parent-element diameters.Under these more general assumptions, stability proofs become more involved than under standard mesh conditions.
- The proposed convergence strategy requires weaker stability conditions on the stabilization form sE(·, ·).
- The analysis covers shape-regular meshes containing edges arbitrarily short relative to their parent elements.
- The approach analyzes both standard and alternative stabilization forms, with the latter yielding error bounds consistent with numerical tests and potentially greater robustness for small edges.
- The classical stabilization is equivalent to the H1 seminorm, with one equivalence constant degrading logarithmically when small edges occur.
- The internal stabilization term s◦E(·, ·) can be dropped without detriment to the stability features of the VEM scheme.
2 The continuous and discrete problems
The paper formulates a diffusion problem and its VEM discretization on polygonal meshes, using computable projection and stabilization forms. It permits weaker edge assumptions and considers alternative, boundary-only stabilization choices.
- 2.1 The continuous problem: The continuous model is a diffusion problem on a polygonal domain with an L2 loading term and symmetric diffusion tensor K.
- 2.2 The virtual element method: The mesh family consists of general simple polygons whose boundaries are subdivided into straight edges, with piecewise-constant diffusion tensors.
- 2.2 The virtual element method: The local virtual element space uses vertex values, edge point values, and internal moments as degrees of freedom.
- 2.2 The virtual element method: The local projector Π∇E maps VE to Pk(E) and is orthogonal with respect to the element bilinear form aE(·, ·).
- 2.2 The virtual element method: The stabilization form sE(·, ·) is symmetric and positive semidefinite, and is split into boundary and internal contributions.
- 2.2 The virtual element method: The paper considers alternative boundary stabilization forms, including one based on tangent derivatives along edges.
- 2.2 The virtual element method: The internal stabilization contribution can be completely neglected without spoiling the numerical scheme’s stability features.
- 2.2 The virtual element method: Assumptions A1 and A2 allow arbitrarily small edges relative to element diameters, whereas A3 imposes a stronger edge requirement.
3 Preliminary results
The preliminary analysis establishes geometric, trace, inverse, approximation, and operator estimates under the mesh assumptions, supporting later VEM stability and convergence arguments.
- Trace estimates: The analysis derives uniform trace and boundary estimates, including a bound for divergence-free normal traces in H−1/2(∂E).For divergence-free w, the normal-trace norm is controlled by the elementwise L2 norm.
- Geometric estimates: Anisotropic scaling and radial mappings transfer standard estimates between elements and comparable reference domains.The construction uses uniformly controlled mappings between each element and its inscribed ball.
- Approximation estimates: Polynomial inverse and approximation estimates extend to polygons satisfying the stated geometric assumptions.The proof uses an inscribed equilateral triangle, bubble-function arguments, anisotropic interpolation, and uniformly bounded maximum angles.
- Operator estimates: The preliminary lemmas establish approximation properties for the operators used in the virtual element construction.Different operator definitions require different regularity assumptions, while additional bounds follow under the second mesh assumption.
4 A general error analysis
This section develops a general VEM error analysis for symmetric elliptic problems by imposing stability conditions on the discrete bilinear form. It then reduces verification of those conditions to boundary stabilization, including a formulation that omits internal degrees of freedom.
- General framework: The approach extends VEM error analysis beyond the standard setting and applies to other linear symmetric elliptic problems.The section explicitly presents the analysis as more general than the standard treatment.
- Stability and well-posedness: The discrete bilinear form is coercive under assumptions (36)–(37), making the discrete problem positive definite with a unique solution.The paper derives positive definiteness from the stability result and the coercivity of the continuous problem.
- Error analysis: The error theorem bounds the discrete solution through an interpolant, a polynomial approximation, and constants governing the stability assumptions.The estimate applies when the continuous solution belongs locally to the virtual element spaces and the comparison polynomial has degree k.
- Stabilization choices: The stabilization analysis considers both the standard internal contribution and a formulation that completely neglects internal degrees of freedom.Both choices are treated with the same boundary bilinear form, whose specific forms are analyzed later.
- Stabilization analysis: A polynomial matching the virtual function's Laplacian provides the key decomposition used to analyze stabilization.For every virtual element function, the paper constructs a degree-k polynomial with the same Laplacian.
- Reduction to the boundary: For the considered internal stabilization choices, boundary control implies the required stability constants C1(E) and C2(E).The propositions establish C1(E) ≲ max{1, bC1(E)} and C2(E) ≲ max{1, bC2(E)}.
5 Analysis of some choices for the boundary stabilization
The paper analyzes several boundary stabilization choices under general mesh assumptions, including meshes with arbitrarily small edges. The resulting stability and convergence bounds identify logarithmic degradation for the identity-matrix choice and uniform bounds for a boundary-derivative stabilization under assumption A1.
- 5.1 Identity matrix choice: The identity-matrix stabilization is the standard, simpler-to-code choice and is associated with an identity matrix of dimension Nk.Its boundary form is analyzed through the edge-based degrees of freedom.
- 5.1 Identity matrix choice: The analysis uses edge-based boundary estimates, including the H1/2_00(Γ) space and an equivalent weighted norm involving distance from the endpoints of Γ.These tools support estimates on the meshed boundary ∂E.
- 5.1 Identity matrix choice: Under assumptions A1 and A2, the identity-matrix boundary form satisfies the stability conditions for the considered choices of operator R.The associated constants satisfy bC1(E) ≲ log(1 + hE/hm(E)) and bC2(E) ≲ 1.
- 5.1 Identity matrix choice: Arbitrarily small edges are allowed when the number of edges is uniformly bounded, with only a logarithmic factor lost in the VEM convergence rate.Under the stronger assumption A3, the factor c(h) is uniformly bounded.
- 5.2 A stabilization based on boundary derivatives: Applying the standard analysis to the boundary-derivative form would produce a strongly suboptimal result in the presence of small edges.The paper attributes this difference to the stronger behavior of the present analysis compared with the standard approach.
- 5.2 A stabilization based on boundary derivatives: For the boundary-derivative stabilization, the new analysis gives stability and convergence bounds under A1 for the considered choices of operator R.The result applies to the boundary form defined in (107).
- 5.2 A stabilization based on boundary derivatives: A variant of the boundary-derivative choice is theoretically robust only under A3 because its bound contains the factor hE/hm(E).The stated factor compares the element diameter with the minimum edge length.
6 Numerical tests
The numerical tests examine VEM solutions on several polygonal mesh families, including meshes with very small edges, and compare classical and boundary stabilizations across polynomial orders.
- Small edges: The classical stabilization produces small oscillations near small edges for k = 1, while they are practically negligible for k = 2.The oscillations are of the order of the approximation error.
- Small edges: The boundary stabilization eliminates the oscillations for k = 1, but its initial choice is too strong and makes the VEM solution less accurate.Using τ = 0.1 improves the situation in the reported experiments.
- Convergence in H1: The convergence tests measure the H1-seminorm error by comparing the exact gradient with the elementwise L2-projection of the VEM gradient onto Pk−1.This projected quantity is computable from the VEM degrees of freedom.
- Convergence in H1: Four mesh sequences are tested: square grids, almost regular hexagons, random Voronoi polygons, and centroidal Voronoi tessellations after 100 Lloyd iterations.The sequences contain progressively refined meshes with different polygonal structures.
- Convergence in H1: As h goes to zero, all tested stabilizations behave similarly, with convergence of order O(hk), as predicted by the theory.The comparison includes the classical stabilization and boundary stabilization with τ = 1 and τ = 0.1, for k = 1 and k = 5.