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A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation

Chunmei Wang, Junping Wang, Ruishu Wang, Ran Zhang

arXiv:1508.03855v1math.NAcs.NA

TL;DR

Nearly incompressible elasticity can exhibit locking, and the locking-free status of the primal WG method required rigorous justification. The paper develops equivalent primal and mixed WG schemes and proves locking-free convergence, supported by numerical tests.

  • Problem

    For nearly incompressible elasticity, locking can impair displacement convergence, and the locking-free nature of the primal WG scheme needed rigorous justification.

    Source: §1 · pp. 2–3

  • Method

    The paper constructs primal and mixed weak Galerkin schemes using discrete weak operators reconstructed through local element problems on polygonal or polyhedral meshes.

    Source: §1 · p. 2, §2 · p. 4, §3 · p. 5

  • Results

    The primal and mixed WG schemes are equivalent, and the paper establishes locking-free convergence theory supported by numerical tests.

    Source: §1 · p. 3, §4 · p. 8, §10.2 · p. 25

  • Takeaways & Limitations

    The primal WG method is presented as accurate and robust on polygonal or polyhedral partitions and locking-free with respect to the Lamé constant λ.

    Source: §1 · p. 2

    The stated estimates assume shape-regular partitions and regularity including w ∈ H^(r+1)(Ω) and ρ ∈ H^r(Ω).

    Source: §3 · p. 5, §9.2 · p. 22

Abstract

from arXiv · show

This paper presents an arbitrary order locking-free numerical scheme for linear elasticity on general polygonal/polyhedral partitions by using weak Galerkin (WG) finite element methods. Like other WG methods, the key idea for the linear elasticity is to introduce discrete weak strain and stress tensors which are defined and computed by solving inexpensive local problems on each element. Such local problems are derived from weak formulations of the corresponding differential operators through integration by parts. Locking-free error estimates of optimal order are derived in a discrete $H^1$-norm and the usual $L^2$-norm for the approximate displacement when the exact solution is smooth. Numerical results are presented to demonstrate the efficiency, accuracy, and the locking-free property of the weak Galerkin finite element method.

1. Introduction.

The paper develops weak Galerkin methods for linear elasticity in primal and mixed formulations, targeting locking-free approximation on polygonal and polyhedral meshes. It establishes the formulations’ equivalence and a locking-free convergence theory, with numerical tests and error analysis for arbitrary polygonal or polyhedral partitions.

1. Introduction.

  • The paper applies WG to the primal elasticity formulation, reconstructing differential operators through inexpensive local element problems derived by integration by parts.Its target includes accuracy and robustness on polygonal and polyhedral partitions, as well as locking-free behavior with respect to the Lamé constant λ.Source: §1 · p. 2
  • For nearly incompressible elasticity, locking can produce poor observed displacement convergence, while existing conforming-method results depend on mesh type and polynomial degree.The paper contrasts this background with its aim of a locking-free WG method.Source: §1 · p. 2
  • A mixed reformulation introduces pressure and can yield locking-free approximations, but requires a saddle-point problem with an additional variable.The paper’s WG primal and mixed formulations are equivalent, motivating its locking-free analysis.Source: §1 · p. 3
  • WG primal and mixed schemes are equivalent, and the paper establishes locking-free convergence theory for the displacement formulation.The analysis addresses shape-regular partitions of arbitrary polygons or polyhedra and uses Korn’s inequality with a domain inverse inequality.Source: §1 · p. 3
  • The paper derives optimal-order displacement error estimates in discrete H1 and L2 norms and presents numerical results demonstrating accuracy and locking-free behavior.The paper organization identifies these estimates and numerical tests as later sections’ contributions.Source: §1 · pp. 3–4

2. Weak Divergence and Weak Gradient Operators.

The method represents a weak function with separate element-interior and boundary values, then defines weak differential operators through integration by parts. Their discrete forms are polynomial reconstructions on each element.

2. Weak Divergence and Weak Gradient Operators.

  • A weak function has interior and boundary values that need not be related by a trace, allowing the method to represent them separately.The discrete weak gradient and divergence are then defined through local inner-product identities.Source: §2 · p. 4
  • The discrete weak divergence is the unique polynomial in Pr(K) whose inner products match the interior divergence term and boundary normal-flux term.Its defining identity tests against every φ in Pr(K).Source: §2 · p. 4
  • The discrete weak gradient is the unique matrix-valued polynomial in [Pr(K)]d×d defined by interior and boundary terms from integration by parts.Its identity tests against every matrix-valued polynomial ϕ in [Pr(K)]d×d.Source: §2 · p. 4
  • The weak strain and stress tensors are defined using the weak gradient, with stress given by σw(u) = 2µεw(u) + λ(∇w · u)I.The paper also describes weak divergence and weak gradient as primary operators in the variational problem.Source: §2 · pp. 4–5

3. Numerical Algorithms.

The numerical algorithm builds a weak finite element space on shape-regular polygonal or polyhedral partitions and computes discrete weak derivatives locally. The resulting primal WG scheme has a unique solution for every λ ≥ 0.

3. Numerical Algorithms.

  • The mesh consists of shape-regular polygons in 2D or polyhedra in 3D, with boundary components shared consistently across interior interfaces.The partition notation distinguishes interior edges or faces from the domain boundary.Source: §3 · p. 5
  • The local weak space pairs interior and boundary components, with rigid-motion traces included in the boundary space.The global space patches local spaces with a single-valued boundary component on each interior interface.Source: §3 · p. 5
  • For each element, the method computes discrete weak divergence and gradient using polynomial tests of degree k−1.The boundary space is [Pk−1(e)]d for k > 1 and the rigid-motion trace space for k = 1.Source: §3 · p. 5
  • Weak Galerkin Algorithm 1 seeks a discrete displacement with prescribed projected boundary values and is formulated through the method’s bilinear forms.The bilinear forms use weak strain and weak divergence computed from discrete weak derivatives.Source: §3 · p. 6
  • The primal WG scheme has a unique solution for every λ ≥ 0.The proof reduces existence to uniqueness for the finite-dimensional system.Source: §3 · pp. 6–7

4. An Equivalent Mixed Formulation.

The paper introduces a mixed weak Galerkin formulation with an auxiliary variable and proves it is equivalent to the primal formulation. This equivalence motivates using mixed-form stability and accuracy analysis to study locking-free convergence of the primal scheme.

4. An Equivalent Mixed Formulation.

  • The mixed formulation assumes H1+s × Hs regularity for some s in (1/2, 1], with estimates stated for smooth data.The cited discussion identifies convex polygonal domains with s = 1 when λ = ∞ as one setting where this regularity estimate is available.Source: §4 · p. 7
  • The mixed formulation adds a finite element space for the auxiliary variable p and seeks displacement and p together.It is presented as a generalized Stokes system with a nonzero divergence constraint.Source: §4 · pp. 7–8
  • The two weak Galerkin algorithms produce identical displacement solutions.The proof eliminates the mixed variable as p̃h = λ∇w · ũh and substitutes it into the displacement equation.Source: §4 · p. 8
  • The equivalence suggests that stability and accuracy of the mixed scheme can establish locking-free convergence for the primal displacement formulation.The paper makes this implication conditional on proving the mixed WG method stable and accurate in terms of λ.Source: §4 · p. 8

5. Stability Conditions.

This section establishes stability tools for the discrete weak Galerkin spaces, including a norm property and an inf-sup condition. These results support the subsequent analysis of the mixed scheme.

5. Stability Conditions.

  • The discrete semi-norm defined in (5.1) is a norm on the zero-boundary finite element space.The proof uses the vanishing strain and boundary projection conditions together with Korn’s inequality.Source: §5 · p. 9
  • The section also relates the discrete semi-norm to elementwise estimates through inequalities (5.5) and (5.6).Summing those estimates over all mesh elements yields the two inequalities in (5.2).Source: §5 · p. 9
  • The section establishes the inf-sup condition (5.7) for the discrete spaces.The proof chooses a specific test function built from the discrete pressure gradient and jumps.Source: §5 · p. 10

6. Preparation for Error Estimates.

The paper prepares its error analysis with projection identities and a consistency relation for sufficiently regular solutions. These results connect projected exact solutions to the weak Galerkin operators.

6. Preparation for Error Estimates.

  • The elementwise projection Q0 maps onto [Pk(T)]d, while Qhu is formed from the element and boundary projections.The passage defines Qhu = {Q0u, Qbu}.Source: §6 · p. 10
  • The projection Qh commutes with the discrete weak strain: εw(Qhv) = Qhε(v).The stated identity holds for v in [H1(Ω)]d.Source: §6 · p. 10
  • For sufficiently regular (w, ρ) satisfying (6.4), the projected pair obeys the discrete identity (6.5), up to functionals ℓw and θρ.The identity uses the projections Qhw and Qhρ and the weak strain and divergence operators.Source: §6 · p. 11

7. Error Estimate in a Discrete H1-Norm.

This section derives error estimates for the mixed weak Galerkin solution under smoothness assumptions. It gives a discrete-norm bound for the pressure error and uses the resulting estimates to obtain the stated error estimates.

7. Error Estimate in a Discrete H1-Norm.

  • For smooth solutions with (u, p) in [Hk+1(Ω)]d × Hk(Ω), Theorem 7.2 derives error estimates for the weak Galerkin solution.The theorem states that its constant is generic and independent of (u, p).Source: §7 · p. 13
  • |||ζh|||0 ≤ C h^k(∥u∥k+1 + ∥p∥k) bounds the pressure error in the stated discrete norm.The bound is given in (7.11).Source: §7 · p. 14
  • The proof combines estimate (7.9) with the pressure-error bound to obtain (7.7), then uses the triangle inequality for (7.8).The latter estimate concerns the L2 error, as indicated in the cited passage.Source: §7 · p. 14

8. Error Estimate in L2.

Under smooth-solution and dual-regularity assumptions, the analysis derives an L2 error estimate for displacement using a duality argument.

8. Error Estimate in L2.

  • A dual problem for the displacement error yields an identity expressing ∥e0∥² as two residual terms.The proof then estimates both terms to obtain the L2 error estimate.Source: §8 · pp. 14–15
  • The proof separately bounds stabilization and consistency contributions before combining them to obtain the estimate.The contributions include s(Qhu, QhΦ), ℓu(QhΦ), and θp(QhΦ).Source: §8 · pp. 16–17
  • Under the stated smoothness and dual-regularity assumptions, the analysis derives an L2 error estimate for displacement.The theorem assumes (u; p) ∈ [Hk+1(Ω)]d × Hk(Ω), k ≥ 1, and regularity assumption (8.4).Source: §8 · pp. 14, 15, 17

9. Supporting Tools and Inequalities.

The section collects inequalities used in the error analysis, including Korn inequalities and technical estimates under shape regularity.

9. Supporting Tools and Inequalities.

  • Trace, inverse, and approximation inequalities are collected for shape-regular partitions and polynomial finite element spaces.The section identifies a trace inequality for element sides, an inverse inequality in polynomial spaces, and estimates for specified Sobolev-regular functions.Source: §9 · p. 17

9.1. Korn’s inequality.

  • Korn’s inequality bounds the gradient by the strain and displacement norms, while zero strain characterizes rigid motions.The stated characterization is for functions in [H1(Ω)]d on an open, bounded, connected domain.Source: §9.1 · pp. 17–18
  • A second Korn inequality supports control after rigid-motion modes are fixed using a subdomain projection or boundary values.The corollaries consider projection onto rigid motions on a subdomain and vanishing values on a nontrivial boundary portion.Source: §9.1 · pp. 18–20

9.2. Some technical inequalities.

  • The local technical tools also relate rigid-motion projections across affine rescaling and provide estimates for discrete functions under regularity conditions.The cited discrete-function estimates assume shape regularity, k ≥ 1, and 1 ≤ r ≤ k.Source: §9.2 · pp. 20, 22
  • On shape-regular elements, an interior ball has radius proportional to element size, enabling a strain-based local estimate.For the ball estimate, the squared L2 norm after subtracting the rigid-motion projection is bounded by C r² times the squared strain norm.Source: §9.2 · p. 21

10. Numerical Results.

The numerical tests examine lowest-order weak Galerkin variants on the unit square. They report the expected convergence orders and locking-free convergence across several norms and parameter settings.

10. Numerical Results.

  • The lowest-order tests compute discrete weak gradient and divergence elementwise, using either rigid-motion or linear boundary spaces.The reported error norms include an H1-like norm and L2 norms for interior and boundary errors.Source: §10 · p. 23
  • Errors are also reported in an H1-like norm and in separate L2 norms for interior and boundary errors.The error is defined from the projection of the exact solution minus the WG solution.Source: §10 · p. 24

10.1. Test Problem 1.

  • On Problem 1, convergence is order 2 in L2 and order 1 in H1, matching theory; Vh and V̄h have the same theoretical order, with Vh using fewer degrees of freedom.The tables compare rigid-motion and linear boundary spaces.Source: §10.1 · p. 24, Table 10.1 · p. 24, Table 10.2 · p. 25

10.2. Test Problem 2.

10.3. Locking-Free Tests.

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