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A posteriori error estimates for a Virtual Elements Method for the Steklov eigenvalue problem
David Mora, Gonzalo Rivera, Rodolfo Rodríguez
TL;DR
The paper addresses a posteriori error estimation for a VEM approximation of the Steklov eigenvalue problem, where flexible polygonal meshes make adaptive refinement attractive. It introduces a residual-type estimator, proves reliability and efficiency, and uses the estimator in adaptive computations. Numerical tests report optimal-order convergence for both regular and singular solutions.
Problem
A posteriori error analysis for VEM is challenging, while flexible polygonal meshes create a need for effective adaptive error indicators for the Steklov eigenvalue problem.
Method
The paper introduces a residual-type a posteriori estimator for the VEM Steklov approximation and uses its indicator to drive adaptive mesh refinement.
Results
The estimator is proved reliable and efficient, and numerical tests confirm optimal-order convergence for regular and singular solutions under adaptive refinement.
Takeaways & Limitations
VEM supports adaptive Steklov computations on general polygonal meshes, including meshes with hanging nodes, while retaining optimal-order convergence in the tested cases.
Takeaways & Limitations
The analysis cannot rigorously establish from the available upper bounds that the boundary L2 error is asymptotically negligible relative to the energy error.
Abstract
from arXiv · showhide
The paper deals with the a posteriori error analysis of a virtual element method for the Steklov eigenvalue problem. The virtual element method has the advantage of using general polygonal meshes, which allows implementing very efficiently mesh refinement strategies. We introduce a residual type a posteriori error estimator and prove its reliability and efficiency. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests, that allow us to assess the performance of this approach.
1. Introduction
The paper develops and analyzes a residual-type a posteriori estimator for a VEM approximation of the Steklov eigenvalue problem, motivated by adaptive refinement on flexible polygonal meshes. It proves estimator reliability and efficiency and uses it to drive an adaptive scheme evaluated numerically.
- Motivation and background: VEM generalizes finite elements to very general polygonal and polyhedral meshes.Such meshes support domain meshing, hanging nodes, moving meshes, and adaptivity.
- Motivation and background: The Steklov eigenvalue problem places the eigenvalue in the boundary condition and has applications including sloshing, structural vibrations, and mechanical stability.
- Motivation and background: A posteriori indicators support adaptive refinement that can achieve errors below a tolerance at reasonable computational cost, including for singular solutions.
- Paper contribution: The paper introduces and analyzes a residual-type estimator for the VEM Steklov approximation, replacing noncomputable normal fluxes by a suitable projection.This replacement produces additional terms representing VEM virtual inconsistency.
- Paper contribution: The estimator is proved equivalent to the error, and its corresponding indicator drives an adaptive scheme.
- Paper organization: The article defines the estimator, proves reliability and efficiency, reports numerical tests, and compares it with the standard finite-element edge-residual estimator.
2. The Steklov eigenvalue problem and its virtual element approximation
The paper formulates the Steklov eigenvalue problem and its VEM discretization on polygonal meshes under geometric regularity assumptions. The discrete method uses computable projected bilinear forms and reproduces the spectral structure, including a zero eigenvalue and positive eigenvalues.
- Problem setting: The domain is a bounded polygonal subset of R2 with boundary split into disjoint open parts Γ0 and Γ1, with Γ0 nonempty.
- Continuous problem: The Steklov problem involves the Laplace operator with the eigenvalue appearing in the boundary condition.
- Continuous problem: The weak formulation is obtained by testing the differential equations with H1 functions and integrating by parts.
- Mesh assumptions: Meshes consist of simple polygons satisfying star-shapedness and minimum vertex-separation conditions uniformly relative to element diameters.
- Virtual element approximation: The VEM uses polygonal local spaces, selected degrees of freedom, broken H1 norms, and bilinear forms that are exactly computable when one argument is a degree-k polynomial.A computable stabilization is included to define the discrete local forms.
- Discrete spectrum and convergence: The discrete spectral problem has λh0 = 0 with constant eigenfunctions and positive eigenvalues repeated according to multiplicity.For a simple positive eigenvalue, normalized discrete eigenfunctions converge in L2(Γ0) and H1(Ω) as h tends to zero.
3. A posteriori error analysis
The paper introduces a fully computable residual-based estimator for the VEM approximation of the Steklov problem and establishes reliability and efficiency. The analysis also addresses computable eigenfunction reconstruction and the asymptotic behavior of a boundary-error term.
- Estimator construction: The estimator is defined from local residual terms, element indicators, and a global estimator computable from the VEM solution.Its construction uses element residuals, edge residuals, and the virtual inconsistency term represented through the stabilization bilinear form.
- Reliability: The residual analysis establishes an upper bound for the energy error through Theorem 3.1.The proof combines an error equation with stability, consistency, trace, and Poincaré estimates.
- Reliability: The computable projection Πhwh can replace the unknown virtual eigenfunction wh while retaining a reliability estimate.The paper explicitly notes that Πhwh is computable and proves a corresponding result in Corollary 3.1.
- Reliability: The boundary error term ∥w − wh∥0,Γ0 cannot initially be shown negligible from the available upper bounds, but Theorem 3.2 establishes its asymptotic negligibility.The refined argument modifies an earlier estimate and proves the needed result for sufficiently small h.
- Efficiency: The local indicators ηK are efficient for identifying polygons that should be refined, with Theorem 3.3 providing the corresponding upper estimate.The efficiency analysis separately bounds residual contributions before deriving the local indicator estimate.
4. Numerical results
Two numerical tests assess a lowest-order VEM adaptive scheme on polygonal meshes for smooth and singular Steklov problems. The reported results show optimal convergence recovery for adaptive VEM and FEM, while the residual estimator remains reliable and efficient.
- Experimental setup: The experiments implement a lowest-order VEM (k = 1) on arbitrary polygonal meshes and compare VEM and FEM refinement procedures.The VEM refinement splits each polygon into quadrilaterals by connecting its barycenter to the vertices; triangular meshes reduce VEM to FEM.
- Test 1: Sloshing in a square domain: For the smooth square-domain sloshing problem, uniform refinement attains the optimal eigenvalue rate, and Figure 5 compares it with adaptive FEM and adaptive VEM.The exact eigenvalues and eigenfunctions are known analytically, with λn = nπ tanh(nπ).
- Test 1: Sloshing in a square domain: The Test 1 estimator has effectivity indexes bounded away from zero and infinity, while inconsistency and edge-residual terms remain of comparable order.Neither estimator component is asymptotically negligible in this test.
- Test 2: The singular-domain test uses a square with an equilateral triangle removed, producing a reentrant angle and reduced eigenfunction regularity.Quasi-uniform meshes are predicted to converge below the optimal rate, motivating adaptive refinement.
- Test 2: The computed lowest eigenvalue in Test 2 is evaluated against λ1 = 1.9288 obtained by least-squares fitting of results from extremely refined meshes.This reference value has at least four correct significant digits.
- Test 2: Adaptive VEM and FEM recover the optimal eigenvalue convergence order in Test 2, whereas uniform refinement follows the lower rate predicted by the singularity.Figure 9 reports error curves for Uniform FEM, Adaptive FEM, and Adaptive VEM, with reference slopes −1 and −3/5.
Conclusions
The paper establishes a reliable and efficient residual a posteriori error indicator for the VEM approximation of the Steklov eigenvalue problem and uses it to drive adaptive refinement. Numerical tests confirm optimal-order convergence for both regular and singular solutions.
- The proposed residual a posteriori error indicator is both reliable and efficient for the VEM Steklov eigenvalue approximation.
- For lowest-order elements on triangular meshes, VEM coincides with FEM, including their a posteriori error indicators.
- VEM supports general polygonal meshes with hanging nodes, enabling the design of an adaptive refinement scheme.
- The adaptive scheme achieves optimal-order convergence for both regular and singular solutions in the reported tests.