Mini mixed finite element method for nearly incompressible linear elasticity problems

1 Introduction

• As ν approaches 0.5 and λ grows, standard low-order conforming methods can fail to approximate the true solution or achieve optimal convergence rates.The paper identifies this as the locking phenomenon.Source: §1 · p. 2
• The Mini element is a low-order mixed element satisfying the inf-sup condition, but its robustness for elasticity as λ grows requires analysis.The paper identifies uniform a priori error analysis for both source and eigenvalue problems as needing detailed exposition and verification.Source: §1 · p. 2
• Efficient, locking-free solvers for elasticity eigenproblems have received less attention than source-problem methods, and standard iterative eigensolvers on locked systems can perform poorly.Adapting multilevel and subspace correction methods to the mixed formulation while ensuring uniform convergence in λ presents distinct challenges.Source: §1 · p. 2
• Multilevel correction methods that reduce eigenvalue computations to linear boundary value solves and a small correction-space eigenproblem inform the proposed solver design.The paper points to these methods as inspiration for addressing the elasticity eigenvalue problem.Source: §1 · p. 3
• The paper develops Mini mixed finite element theory for the boundary value and eigenvalue problems, then proposes an augmented subspace algorithm for eigenproblems in non-nested mixed spaces.The analysis and numerical examples support convergence uniform in λ as λ→∞.Source: §1 · p. 3

2.1 Well-posedness of mixed variational form

• The mixed variational problem has a unique solution with a stability bound whose constant is independent of the Lamé constant λ.The analysis establishes boundedness, an inf-sup condition, and symmetry before applying the Babuška-Lax-Milgram theorem.Source: §2.1 · pp. 4, 6

2.2 Mini mixed finite element method and a priori error estimates

• The Mini mixed finite element error analysis hinges on a bounded linear operator satisfying the required Fortin condition, illustrated for the two-dimensional Mini element.The interpolation operator is used to establish the stability and convergence properties of the discrete spaces.Source: §2.2 · pp. 7–8
• The Mini discretization has a locking-free a priori error estimate with constants independent of λ and h; optimal order follows when u∈(H2(Ω))^d and p∈H1(Ω).The error estimate covers displacement in the V norm and pressure in the Q norm.Source: §2.2 · pp. 7–8

3.1 Problem formulation

• The Mini mixed finite element method discretizes the mixed elasticity eigenvalue problem on the finite-dimensional space Xh = Vh × Qh.The continuous formulation uses displacement and pressure variables, with the displacement normalized in the mass form.Source: §3.1 · pp. 8–9
• The discrete eigenvalues satisfy the upper-bound property ¯λi,h ≥ λi for each i through the dimension Nh of Xh.The passage attributes this property to the max-min principle.Source: §3.1 · p. 9

3.2 Error estimates for eigenfunction

• The projection-operator estimates include constants independent of the Lamé constant λ.The finite element projection estimate is also stated with a constant independent of λ and mesh size h.Source: §3.1 · p. 10, §3.2 · p. 12
• The eigenfunction analysis bounds spectral projection errors and gives an L2 error estimate, with an eigenvalue-gap condition in the general estimate.The simplified estimate assumes a mesh-independent lower bound on the gap when the mesh is sufficiently fine.Source: §3.2 · pp. 12, 14

4 Augmented subspace method based on Mini mixed finite element

• The method specifically addresses locking in the linear elasticity eigenvalue problem.Its non-nested construction is designed for the mixed finite element setting.Source: §4 · p. 16
• The augmented space combines coarse Mini functions with a fine-space function even though the coarse and fine spaces are not nested.The paper states that the error estimates for the original mixed discretization remain valid in this augmented space.Source: §4 · pp. 16–17
• The augmented subspace iteration updates the space and solves an eigenvalue problem repeatedly until convergence.The algorithm also uses a linear boundary value problem as part of its iteration.Source: §4 · pp. 16–17
• The analysis establishes bounds for the augmented-space approximation with constants independent of the Lamé constant λ.The error estimates for the augmented subspace are stated to remain valid, and the operator bound is λ-independent.Source: §4 · p. 17

5 Numerical experiments

• The numerical tests use high-performance computers at the State Key Laboratory of Mathematical Sciences, Chinese Academy of Sciences.The passage identifies these computers as the computational platform for the tests.Source: §5 · p. 18
• Each computer node has two 18-core, 2.3 GHz Intel Xeon Gold 6140 processors and 192 GB of RAM.These hardware specifications are reported for each node.Source: §5 · p. 18
• The section presents several numerical tests to support the theoretical findings.The passage does not specify their results or comparisons.Source: §5 · p. 18

5.1 Numerical tests for boundary value problem

• The boundary-value experiments comprise a convergence-order test and a test of the locking-free property.The section introduces two numerical examples to validate the theoretical results.Source: §5.1 · p. 18, §5.1.1 · p. 18, §5.1.2 · p. 19

5.1.1 Numerical test for convergence order

• About 1 for displacement in the H1-norm and 1.5 for pressure in the L2-norm are the reported convergence orders.The paper describes the displacement order as optimal and the pressure order as superconvergent.Source: §5.1.1 · p. 19, Table 1 · p. 19

5.1.2 Numerical test for locking-free property

• As λ increases significantly, displacement H1 and pressure L2 errors vary within a small range, indicating locking-free behavior.The test spans λ = 1, 10, 10^3, and 10^6.Source: §5.1.2 · p. 19

5.2 Numerical tests for eigenvalue problem

• The tests use λ = 10, 10^3, and 10^6 to compare the three elements on minimum and fourth eigenvalues.The problem uses a two-dimensional unit square with homogeneous displacement boundary conditions and a quasi-uniform triangular mesh.Source: §5.2 · p. 19, Table 3 · p. 20
• As λ increases, Mini mixed and CR eigenvalues remain stable, while linear conforming eigenvalues become large.The comparison covers the minimum and fourth eigenvalues.Source: §5.2 · pp. 19–20, Table 3 · p. 20
• The linear conforming element’s eigenvalues deteriorate as λ tends to infinity, unlike the locking-free Mini mixed and CR results.Source: §5.2 · p. 20

5.3.1 The minimum eigenvalue corresponds to the convergence of eigenfunction

• For the first eigenfunction, Algorithm 1 has second-order convergence with respect to H, and errors and rates remain nearly unchanged as λ rises from 10 to 10^6.The figures cover λ = 10, 10^3, and 10^6.Source: §5.3.1 · p. 21, Figure 1 · p. 21, Figure 2 · p. 22

5.3.2 The first four eigenvalues correspond to the convergence of eigenfunction

• For the smallest four eigenfunctions, the fourth-mode example converges at second order in H, more slowly than the first mode; errors and rates stay nearly unchanged as λ rises to 10^6.The fourth-mode convergence rates reported are 0.22751, 0.066263, and 0.016969.Source: §5.3.2 · p. 22, Figure 4 · p. 23, Figure 5 · p. 23

5.3.3 The fourth eigenvalue corresponds to the convergence of eigenfunction

• For the fourth eigenfunction alone, Algorithm 1 converges at second order in H, faster for smaller coarse meshes, while errors and rates remain largely unchanged as λ rises to 10^6.The reported rates are 0.22988, 0.065701, and 0.015915.Source: §5.3.3 · pp. 22, 24, Figure 7 · p. 24, Figure 8 · p. 24

• Error estimates independent of material parameters verify the locking-free property of the Mini mixed finite element discrete solution.Source: §6 · p. 25
• For eigenvalue problems, the paper proposes and analyzes an augmented subspace method within the mixed finite element framework.Source: §6 · p. 25
• Numerical experiments verify the algorithm's algebraic error order and locking-free property.Source: §6 · p. 25
• The study covers almost incompressible linear elasticity source and eigenvalue problems.Source: §6 · p. 25