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A posteriori error estimates for the virtual element method
Andrea Cangiani, Emmanuil H. Georgoulis, Tristan Pryer, Oliver J. Sutton
TL;DR
The paper addresses the difficulty of obtaining fully computable a posteriori error bounds for VEM. It develops a residual-type estimator using projected fluxes and VEM interpolation tools, proves upper and lower bounds, and applies the estimator to adaptive refinement on flexible meshes. The analysis supports optimal convergence in the reported test problems, while a symmetric coefficient configuration can cause repeated effectivity problems.
Problem
Fully computable a posteriori error bounds are difficult for VEM because virtual basis-function normal fluxes are unavailable, and prior analysis was limited.
Method
The paper replaces unavailable normal fluxes with projected fluxes, adds virtual inconsistency terms, and constructs a two- and three-dimensional Clément-type VEM interpolant.
Results
The estimator has lower bounds for inconsistency terms and supports adaptive refinement with optimal H1-seminorm rates of N^-p/2 and N^-1/2 in reported test problems.
Takeaways & Limitations
VEM adaptive refinement can use very general polygonal/polyhedral meshes, including co-planar interfaces and hanging nodes, without local mesh post-processing.
Takeaways & Limitations
Symmetric refinement configurations can make coefficient-oscillation terms jump and cause problems with estimator effectivity.
Abstract
from arXiv · showhide
An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing.
1. Introduction.
The paper develops computable residual-based a posteriori error bounds for C0-conforming VEM on general polygonal/polyhedral meshes, addressing limitations in the sparse existing analysis. Its estimator supports adaptive refinement without local mesh post-processing.
- Fully computable error analysis is challenging because standard VEM normal fluxes are unavailable.
- The analysis targets C0-conforming VEM for second-order linear elliptic reaction-convection-diffusion problems with variable coefficients in two and three dimensions.
- Projected fluxes replace unavailable VEM normal fluxes, with virtual inconsistency terms accounting for the replacement error.
- A new VEM Clément-type interpolant enables minimal-regularity interpolation using linear VEM functions.
- General polygonal/polyhedral meshes allow hanging nodes and co-planar interfaces without local post-processing during refinement.
- The estimator is used in automatic adaptive mesh refinement experiments, with numerical results confirming optimality.
2. The Continuous Problem.
The continuous problem is a second-order elliptic boundary-value problem with diffusion, convection, reaction, and forcing terms. The formulation splits its bilinear form into symmetric and skew-symmetric components to preserve discrete coercivity.
- The model assumes bounded reaction and forcing data, a bounded convection field, and a strongly elliptic symmetric diffusion tensor.
- The variational problem combines diffusion, convection, and reaction terms and equates them with the forcing functional.
- The bilinear form is rewritten as A(u,v) := a(u,v) + b(u,v), separating symmetric and skew-symmetric parts.
- This splitting preserves coercivity of the discrete operator independently of mesh size.
- An alternative formulation without coercivity uses sufficiently small mesh sizes and admits the same a posteriori analysis after minor modifications.
3. The Virtual Element Method.
The VEM constructs conforming virtual spaces on polygonal and polyhedral meshes using degrees of freedom and computable polynomial projections. Its formulation preserves polynomial components while stabilizing the non-polynomial virtual part.
- Mesh assumptions: Meshes may contain non-convex elements and consecutive co-planar edges or faces, subject to star-shapedness and interface-size regularity assumptions.
- Mesh assumptions: A shape-regular sub-triangulation exists for each polygonal or polyhedral element, supporting analysis and construction.
- Virtual spaces: The local space contains Pp(E), while its virtual functions may also include non-polynomial components.
- Computability: VEM terms are computable when evaluated from problem data, degrees of freedom, and the polynomial component of the virtual space.
- Degrees of freedom: Degrees of freedom consist of nodal values and polynomial moments defined on vertices, edges, faces, and element interiors.
- Computable projections: Higher-order moments and projection operators make the elementwise L2 projection computable from a reduced degree-of-freedom set.
- Discrete formulation: The discrete problem has a unique solution under the stated coercivity condition and optimal-order H1 and L2 a priori error bounds.
4. Approximation properties.
The approximation analysis establishes interpolation and inverse estimates for virtual element functions on polygonal and polyhedral meshes. The construction uses classical Clément interpolation on regular sub-triangulations and extends it across faces in three dimensions.
- Proof ingredients: The proof uses bubble functions, trace estimates, Sobolev interpolation, and Poincaré inequalities to control interior and interface contributions.
- Inverse estimates: Virtual element functions admit an inverse estimate when their Laplacians are elementwise polynomials of degree p.
- Interpolation: The new Clément-type construction is designed to generalize approximation results to three dimensions.
- Interpolation: A classical Clément interpolant on a globally shape-regular sub-triangulation provides the starting approximation.
- Interpolation: Theorem 5 constructs a virtual interpolant vI in the VEM space with elementwise approximation estimates for functions in H1(Ω).
- Three-dimensional construction: In three dimensions, facewise two-dimensional interpolants are combined and extended into each polyhedral element.
5. A posteriori error analysis.
The analysis derives computable residual-based a posteriori bounds for C0-conforming VEM by replacing unavailable normal fluxes with projected fluxes and accounting for the resulting virtual inconsistency. It proves upper and local lower bounds, including projected-solution bounds, and tests estimator-driven adaptivity numerically.
- Estimator construction: The residual equation contains additional virtual inconsistency terms because the VEM bilinear form and solution differ from their continuous counterparts.These terms arise explicitly in the error identity before each contribution is bounded.
- Upper bound: Theorem 1 establishes a global upper bound for the VEM energy-norm error with a constant independent of h, u, and uh.The constant depends on mesh, domain, stability, and PDE-coefficient parameters rather than the discretisation size or solutions.
- Estimator construction: The computable estimator combines element and edge residuals, data oscillation, virtual inconsistency terms, and a stabilisation term based on the polynomial part of the VEM solution.The residual quantities are computable because they use the polynomial projection rather than the unavailable virtual solution itself.
- Lower bounds: The estimator also bounds the error between the exact solution and the projected virtual element solution, while the virtual inconsistency terms are controlled up to data oscillation.These results establish lower-order control for both the projected solution error and the inconsistency contributions.
- Lower bounds: A local lower bound is proved for the residual, stabilisation, and data-oscillation contributions using element and edge bubble functions.The local estimate yields a corresponding global lower bound by summing over the mesh.
- Numerical validation: Numerical experiments apply the estimator in an adaptive algorithm to varied test problems, confirming numerically its optimality.The paper uses the estimator to test practical behaviour and drive mesh adaptivity.
6. Numerical Results.
The numerical experiments show optimal convergence and effective error estimation for uniform and adaptive VEM computations on polygonal meshes, including challenging coefficient jumps.
- Uniformly generated meshes: Uniformly refined non-convex polygonal meshes achieve optimal convergence rates for both the H1(Ω)-seminorm error and estimator at p = 1, 2, and 3.Asymptotic effectivities approach approximately 5.7, 3, and 1.84 for p = 1, 2, and 3, respectively.
- Adaptive refinement: The adaptive algorithm follows a solve→estimate→mark→refine cycle using Dörfler marking to select elements with the largest estimated errors.The reported implementation uses θ = 0.4 for the marking parameter.
- Adaptive refinement: Adaptive refinement reaches the theoretical optimal rate N^−p/2 after the asymptotic regime, despite low regularity near the reentrant corner in Problem 1.For initially unresolved features, data oscillation dominates early refinement before element and face residuals become dominant.
- Adaptive refinement: For both adaptive test problems, effectivity plots show good agreement between the estimated and calculated errors.The estimator components change during refinement as localized features become resolved.
- Jumping diffusion coefficient: For jumping diffusion coefficients, square and randomized quadrilateral meshes reach the theoretical N^−1/2 rate, while Voronoi meshes achieve approximately N^−0.35.Estimator convergence reflects these rates, with roughly constant effectivities on the Voronoi and randomized quadrilateral meshes.
- Conclusions: The numerical analysis concludes that the residual estimator is equivalent to the VEM energy error and supports adaptive computations on general elliptic problems.The conclusion attributes the analysis to a new Clément-type interpolation result and reports extensive numerical validation.
7. Conclusions and extensions.
The a posteriori analysis extends to related virtual element formulations with only minor modifications. For a formulation that discretizes the problem directly, the resulting estimator omits the ΨE term.
- The a posteriori analysis applies to other related virtual element formulations with only minor modifications.
- Directly discretizing the problem without splitting the differential operator yields the same a posteriori estimator without the ΨE term.