Source-linked AI summary
Partition of unity a posteriori error estimators for the linear elasticity eigenvalue problem
Joscha Gedicke, Karen Petersen
TL;DR
The paper examines whether partition-of-unity a posteriori estimators are reliable and efficient for linear-elasticity eigenvalue problems with different boundary conditions. It analyzes local residual problems and proves reliability and local efficiency for the estimators.
Problem
The paper investigates the reliability and efficiency of a posteriori error estimators for linear-elasticity eigenproblems with Dirichlet, Neumann, and gliding boundary conditions.
Source: §1 · p. 1Method
The estimators use a partition of unity to localize residuals and solve local problems in displacement formulation.
Source: §1 · p. 2Results
Both estimators are globally reliable for the continuous and discrete eigenpairs, and both are locally efficient.
Source: §1 · p. 3Takeaways & Limitations
The paper establishes reliability and local-efficiency results for these estimators in the eigenvalue setting.
Source: §1 · p. 3The local problems require polynomial degree k ≥ 2 to ensure residual orthogonality to rigid body motions.
Source: §3.4 · p. 16
Abstract
from arXiv · showhide
In this paper we consider the linear elasticity eigenvalue problem with different boundary conditions and examine two a posteriori error estimators, that compute solutions to local residual problems using a partition of unity and localized residuals. These a posteriori error estimators are known for the source problem, and in this paper we investigate their reliability and efficiency for the given eigenvalue problem with Dirichlet, Neumann and gliding boundary conditions. In numerical experiments we test the performance of the estimators on different domains, with different boundary conditions, and with different polynomial degrees. The numerical experiments show efficiency indices closer to one compared to the standard residual-based a posteriori error estimator.
1 Introduction
The paper studies partition-of-unity a posteriori estimators for linear-elasticity eigenproblems with Dirichlet, Neumann, and gliding boundary conditions. It establishes reliability and efficiency properties for the estimators and describes their local-problem formulation.
1 Introduction
- The problem allows Dirichlet, Neumann, and gliding boundary parts, with gliding conditions imposing zero normal displacement and zero tangential traction.Each boundary part is connected, and up to two may be empty.Source: §1 · p. 1
- The estimators solve local residual problems in displacement formulation, using a partition of unity to localize residuals.This formulation may require finite elements of higher polynomial degree, but the paper reports that increasing the degree by one sufficed in its numerical experiments.Source: §1 · p. 2
- The approach extends prior partition-of-unity estimators for linear-elasticity boundary-value problems to eigenvalue problems.The cited prior estimator work considered the source problem; other cited eigenvalue estimators addressed settings including Dirichlet conditions and mixed formulations.Source: §1 · p. 2
- Both estimators are globally reliable for the continuous and discrete eigenpairs, and each is also locally efficient.The reliability bounds are stated with a constant depending only on the domain.Source: §1 · p. 3
2 Weak formulation
The weak formulation places the Dirichlet and normal-displacement constraints in the solution space, while the Neumann and gliding conditions make the boundary integral vanish. The discrete problem uses piecewise-polynomial spaces.
2 Weak formulation
- The solution space H imposes zero displacement on ΓD and zero normal displacement on Γg.The weak formulation is posed for normalized eigenfunctions in this space.Source: §2 · pp. 3–4
- The boundary integral vanishes across all three boundary parts: Neumann traction is zero, Dirichlet data are built into H, and gliding conditions impose contact without friction.On Γg, the contact condition is zero normal displacement and the no-friction condition is zero tangential traction.Source: §2 · pp. 3–4
- The discrete eigenproblem follows the continuous weak formulation using piecewise-polynomial displacement functions.The polynomial degree is at most k on each triangle.Source: §2 · p. 4
- The discrete eigenvalues converge at twice the rate of the eigenfunctions, and the Rayleigh–Ritz principle gives κh ≥ κ.The stated a priori convergence applies when the solution has regularity parameter s > 1.Source: §2 · p. 5
3 A posteriori error analysis
The section fixes mesh and triangulation notation used in the analysis.
3 A posteriori error analysis
- The analysis assumes a regular, quasi-uniform triangulation and uses h for the mesh-size bound, hT and hE for element and edge sizes, and ≲ and ≈ for bounds up to mesh-independent constants.The notation distinguishes interior, Neumann, Dirichlet, and gliding edges and nodes by subscripts.Source: §3 · p. 5
3.1 Residual for the eigenvalue problem
The section defines a residual for the discrete eigenpair, localizes it with partition-of-unity hat functions, and relates residual norms to the eigenfunction error.
3.1 Residual for the eigenvalue problem
- The residual associated with the discrete eigenpair is a linear functional that vanishes on discrete test functions.Integration by parts rewrites it in terms of element and boundary contributions.Source: §3.1 · p. 5
- The gliding-edge residual contribution uses tangential traction, reflecting the weak incorporation of the no-friction condition.The contact condition removes the normal component of the test function on those edges.Source: §3.1 · p. 6
- A proposition expresses the energy norm of the eigenfunction error in terms of the continuous and discrete eigenpairs.The section also introduces dual residual norms and Korn’s inequality to connect residual norms with the error.Source: §3.1 · p. 6
- Hat functions localize the residual to vertex patches, and summing the local contributions yields global estimators.Each local residual is defined using the hat function associated with its patch.Source: §3.1 · p. 7
3.2 Error estimator η
The section defines η from localized residual contributions and establishes its reliability and efficiency, including bounds for the eigenvalue error.
3.2 Error estimator η
- η is based on the dual norm of the residual and is assembled by summing local estimators over patches.The section analyzes reliability and efficiency for this estimator.Source: §3.2 · p. 7, §3.2.1 · p. 8
3.2.1 Reliability of η
- η is reliable: the estimator bounds the error from above.The proof localizes the residual using the partition of unity and combines the local contributions, with patch overlap bounded by 3.Source: §3.2.1 · p. 8
3.2.2 Efficiency of η
- η is globally efficient, with the remaining terms in its bound being higher order.The global estimate follows from local efficiency together with a bound on patch overlap.Source: §3.2.2 · pp. 9–10
- η is shown to be both reliable and efficient with respect to the eigenvalue error.The corollary states these bounds under the assumptions of the reliability and efficiency theorems.Source: §3.2.2 · p. 11
3.3 Error estimator µ
The section introduces µ as a patch-based residual estimator, proves reliability and efficiency, and relates both bounds to the eigenvalue error.
3.3 Error estimator µ
- Unlike η, µ’s reliability does not rely on a global Korn constant.The section also establishes reliability for µ.Source: §3.3 · p. 11, §3.3.1 · p. 11
3.3.1 Reliability of µ
- µ is reliable and bounds the error from above.The proof localizes the residual and estimates its local contributions using µz.Source: §3.3.1 · pp. 11–12
3.3.2 Efficiency of µ
- µ is efficient, with the remaining terms in its bound being of higher order.The global efficiency estimate is obtained from local bounds.Source: §3.3.2 · pp. 13–14
- µ is shown to be both reliable and efficient with respect to the eigenvalue error.These bounds hold under the assumptions of the reliability and efficiency theorems.Source: §3.3.2 · p. 14
3.4 Computation of ηz and µz
The estimators’ local suprema are approximated by solving discrete problems on patches, with higher-degree local finite elements used for the approximation.
3.4 Computation of ηz and µz
- The local suprema defining ηz and µz are approximated by solving discrete local problems on patches.The local problems use Pk+ℓ finite elements, with ℓ≥1, when the eigenvalue problem uses degree k.Source: §3.4 · pp. 15–16
- The discrete ηz problem is well-posed because the residual vanishes on constants.The section relates its solution to a discretization of the local supremum defining ηz.Source: §3.4 · p. 15
- For µz, rigid body motions require k≥2 to ensure the residual vanishes on the relevant test functions.For the lowest-order case, the authors use η instead.Source: §3.4 · p. 16
4 Numerical experiments
The experiments examine patch constraints and estimator performance across different domains, boundary conditions, and polynomial degrees. They report that adaptive refinement can restore optimal convergence in selected tests and that µ2’s efficiency index is closer to 1 than the residual estimator’s on the L-shaped domain and Cook’s membrane.
4.1 Requirements on different patches
- Patch constraints depend on boundary type and patch location: interior and Neumann patches require rigid-motion conditions, while Dirichlet edges and gliding corners can remove the need for them.For axis-parallel gliding edges, the integral constraints are reduced according to the edge orientation.Source: §4.1 · pp. 16–18
4.2 Experiments on the unit square
- On the Neumann unit square, the efficiency index stabilizes when local problems use one polynomial degree more than the finite elements for η and µ.The experiments use P1-FE for η and P2-FE for µ, then use Pk+1-FE for local problems in subsequent tests unless stated otherwise.Source: §4.2 · p. 18, Figure 4 · p. 19
- On the unit square, adaptive refinement that breaks the criss structure also breaks locking, while structured criss meshes show locking in both error and estimators.The estimator preasymptotic convergence rates worsen by one degree more than the eigenvalue error; using P5-FE for local problems does not change this behavior.Source: Figure 5 · p. 19, Figure 6 · p. 20, §4.2 · p. 20
- For Dirichlet unit-square tests, κ1, κ2, and κ4 do not attain optimal convergence rates because their eigenfunctions are not smooth, and absolute eigenvalue error increases for higher eigenvalues.The computed indices include m = 1,...,4,12,13.Source: §4.2 · p. 20, Figure 7 · p. 21
4.3 Experiments on an L-shaped domain; 4.4 Experiments on Cook’s membrane
- By two orders of magnitude, µ2’s efficiency constant is closer to 1 than η2res on the L-shaped domain; Cook’s membrane shows the same comparison.The experiments compare efficiency indices on the L-shaped domain and Cook’s membrane.Source: Figure 9 · p. 22, §4.3 · p. 22, §4.4 · p. 22
4.3 Experiments on an L-shaped domain
- For the smallest nonzero eigenvalue κ4, P3-FE attains the optimal convergence rate on adaptively refined meshes, but not on uniformly refined meshes.The first three eigenvalues corresponding to rigid body motions are zero.Source: §4.3 · p. 22
4.4 Experiments on Cook’s membrane
- On Cook’s membrane, adaptive refinement attains optimal convergence rates, whereas uniform refinement does not.The test uses mixed Dirichlet and Neumann boundary parts and computes the smallest eigenvalue κ1.Source: §4.4 · p. 22, Figure 11 · p. 23, Figure 10 · p. 23