New hybridized mixed methods for linear elasticity and optimal multilevel solvers

Abstract

In this paper, we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions \(n=2,3\), which yields a conforming and strongly symmetric approximation for stress. Applying \(\mathcal {P}_{k+1}-\mathcal {P}_k\) as the local approximation for the stress and displacement, the mixed methods achieve the optimal order of convergence for both the stress and displacement when \(k \ge n\). For the lower order case \((n-2\le k<n)\), the stability and convergence still hold on some special grids. The proposed mixed methods are efficiently implemented by hybridization, which imposes the inter-element normal continuity of the stress by a Lagrange multiplier. Then, we develop and analyze multilevel solvers for the Schur complement of the hybridized system in the two dimensional case. Provided that no nearly singular vertex on the grids, the proposed solvers are proved to be uniformly convergent with respect to both the grid size and Poisson’s ratio. Numerical experiments are provided to validate our theoretical results.

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References

  1. 1.

    Adams, S., Cockburn, B.: A mixed finite element method for elasticity in three dimensions. J. Sci. Comput. 25(3), 515–521 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Amara, M., Thomas, J.M.: Equilibrium finite elements for the linear elastic problem. Numer. Math. 33(4), 367–383 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Antonietti, P.F., Verani, M., Zikatanov, L.: A two-level method for mimetic finite difference discretizations of elliptic problems. Comput. Math. Appl. 70(11), 2674–2687 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Arnold, D.N., Awanou, G.: Rectangular mixed finite elements for elasticity. Math. Models Methods Appl. Sci. 15(09), 1417–1429 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Arnold, D.N., Awanou, G., Winther, R.: Finite elements for symmetric tensors in three dimensions. Math. Comput. 77(263), 1229–1251 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO-Modél. Math. et Anal. Numér. 19(1), 7–32 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Arnold, D.N., Douglas Jr., J., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45(1), 1–22 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Arnold, D.N., Falk, R., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76(260), 1699–1723 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15(1), 1–155 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Arnold, D.N., Qin, J.: Quadratic velocity/linear pressure stokes elements. Adv. Comput. Methods Partial Differ. Equ. 7, 28–34 (1992)

    Google Scholar 

  11. 11.

    Arnold, D.N., Winther, R.: Mixed finite elements for elasticity. Numer. Math. 92(3), 401–419 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Arnold, D.N., Winther, R.: Nonconforming mixed elements for elasticity. Math. Models Methods Appl. Sci. 13(03), 295–307 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Awanou, G.: A rotated nonconforming rectangular mixed element for elasticity. Calcolo 46(1), 49–60 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Boffi, D., Brezzi, F., Fortin, M.: Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8(1), 95–121 (2009)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer, London (2007)

    Google Scholar 

  16. 16.

    Brenner, S.C.: Multigrid methods for parameter dependent problems. RAIRO-Modél. Math. et Anal. Numér. 30(3), 265–297 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, vol. 15. Springer, Berlin (2012)

    Google Scholar 

  19. 19.

    Chen, L.: iFEM: An Innovative Finite Element Methods Package in MATLAB. University of Maryland, Preprint (2008)

  20. 20.

    Chen, L., Hu, J., Huang, X.: Fast Auxiliary Space Preconditioner for Linear Elasticity in Mixed Form. Mathematics of Computation (2017)

  21. 21.

    Cho, D., Xu, J., Zikatanov, L.: New estimates for the rate of convergence of the method of subspace corrections. Numer. Math.: Theory Methods Appl. 1(1), 44–56 (2008)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Cockburn, B., Dubois, O., Gopalakrishnan, J., Tan, S.: Multigrid for an HDG method. IMA J. Numer. Anal. 34(4), 1386–1425 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comput. 79(271), 1331–1349 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    de Dios, B.A., Georgiev, I., Kraus, J., Zikatanov, L.: A subspace correction method for discontinuous galerkin discretizations of linear elasticity equations. ESAIM: Math. Model. Numer. Anal. 47(5), 1315–1333 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Gopalakrishnan, J.: A Schwarz preconditioner for a hybridized mixed method. Comput. Methods Appl. Math. Comput. Methods Appl. Math. 3(1), 116–134 (2003)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Gopalakrishnan, J., Guzmán, J.: Symmetric nonconforming mixed finite elements for linear elasticity. SIAM J. Numer. Anal. 49(4), 1504–1520 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Gopalakrishnan, J., Tan, S.: A convergent multigrid cycle for the hybridized mixed method. Numer. Linear Algebra Appl. 16(9), 689–714 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Guzmán, J.: A unified analysis of several mixed methods for elasticity with weak stress symmetry. J. Sci. Comput. 44(2), 156–169 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Hong, Q., Kraus, J., Xu, J., Zikatanov, L.: A robust multigrid method for discontinuous Galerkin discretizations of stokes and linear elasticity equations. Numer. Math. 132(1), 23–49 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Hu, J.: Finite element approximations of symmetric tensors on simplicial grids in \(\mathbb{R}^n\): the higher order case. J. Comput. Math. 33(3), 283–296 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Hu, J.: A new family of efficient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation. SIAM J. Numer. Anal. 53(3), 1438–1463 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Hu, J., Shi, Z.C.: Lower order rectangular nonconforming mixed finite elements for plane elasticity. SIAM J. Numer. Anal. 46(1), 88–102 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Hu, J., Zhang, S.: A family of conforming mixed finite elements for linear elasticity on triangular grids. arXiv preprint arXiv:1406.7457 (2014)

  35. 35.

    Hu, J., Zhang, S.: A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci. China Math. 58(2), 297–307 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Hu, J., Zhang, S.: Finite element approximations of symmetric tensors on simplicial grids in \(\mathbb{R}^n\): the lower order case. Math. Models Methods Appl. Sci. 26(09), 1649–1669 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Hu, X., Wu, S., Wu, X.H., Xu, J., Zhang, C.S., Zhang, S., Zikatanov, L.: Combined preconditioning with applications in reservoir simulation. Multiscale Model. Simul. 11(2), 507–521 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Johnson, C., Mercier, B.: Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math. 30(1), 103–116 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Lee, Y.J., Wu, J., Chen, J.: Robust multigrid method for the planar linear elasticity problems. Numer. Math. 113(3), 473–496 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Li, B., Xie, X.: Analysis of a family of HDG methods for second order elliptic problems. J. Comput. Appl. Math. 307, 37–51 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Li, B., Xie, X.: BPX preconditioner for nonstandard finite element methods for diffusion problems. SIAM J. Numer. Anal. 54(2), 1147–1168 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Man, H.Y., Hu, J., Shi, Z.C.: Lower order rectangular nonconforming mixed finite element for the three-dimensional elasticity problem. Math. Models Methods Appl. Sci. 19(01), 51–65 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Morgan, J., Scott, R.: A nodal basis for C\(^1\) piecewise polynomials of degree \(n\ge 5\). Math. Comput. 29(131), 736–740 (1975)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Morley, M.E.: A family of mixed finite elements for linear elasticity. Numer. Math. 55(6), 633–666 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Qiu, W., Demkowicz, L.: Mixed hp-finite element method for linear elasticity with weakly imposed symmetry. Comput. Methods Appl. Mech. Eng. 198(47), 3682–3701 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Qiu, W., Shen, J., Shi, K.: An HDG method for linear elasticity with strong symmetric stresses. Math. Comput. 87(309), 69–93 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Schöberl, J.: Multigrid methods for a parameter dependent problem in primal variables. Numer. Math. 84(1), 97–119 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Schöberl, J.: Robust multigrid methods for parameter dependent problems. Ph.D dissertation, Johannes Kepler Universität Linz (1999)

  49. 49.

    Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO-Modél. Math. et Anal. Numér. 19(1), 111–143 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Soon, S.C., Cockburn, B., Stolarski, H.K.: A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Eng. 80(8), 1058–1092 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Toselli, A., Widlund, O.: Domain Decomposition Methods: Algorithms and Theory, vol. 34. Springer, London (2005)

    Google Scholar 

  53. 53.

    Wu, S., Gong, S., Xu, J.: Interior penalty mixed finite element methods of any order in any dimension for linear elasticity with strongly symmetric stress tensor. Math. Models Methods Appl. Sci. 27(14), 2711–2743 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Yi, S.Y.: Nonconforming mixed finite element methods for linear elasticity using rectangular elements in two and three dimensions. Calcolo 42(2), 115–133 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Yi, S.Y.: A new nonconforming mixed finite element method for linear elasticity. Math. Models Methods Appl. Sci. 16(07), 979–999 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74(250), 543–554 (2005)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

The work of the first and third authors was supported in part by National Natural Science Foundation of China (NSFC) (Grant Nos. 91430215, 41390452) and by Beijing International Center for Mathematical Research of Peking University, China. The work of the second and third authors was supported in part by the DOE Grant DE-SC0009249 as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials and by DOE Grant DE-SC0014400 and NSF Grant DMS-1522615.

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Correspondence to Jinchao Xu.

Appendices

Appendix A: Proof of Lemma 3 (construction of the bases of \(\mathbb {R}(\mathcal {C})\) and \(\mathbb {R}(\mathcal {C}^{\perp })\) )

Proof

Denote the set of all \(k+1\) degree Lagrange nodes in \(\mathcal {T}_h\) by \(A_{h,k+1}\). For any \(K \in \mathcal {T}_h\) and \(a \in A_{h,k+1} \cap \bar{K}\), let \(\varphi _a^K\) be the Lagrange nodal basis in K, with zero extension in \(\mathcal {T}_h {\setminus } K\). Further, for any \(F \in \mathcal {F}_h^i\) and \(a \in A_{h,k+1} \cap \bar{F}\), let \(\psi _a^F\) be the dual basis of the degree \(k+1\) Lagrange basis such that

$$\begin{aligned} \langle \psi _{a'}^F, \varphi _a^K \rangle _F = \delta _{a,a'} \quad \text{ and } \quad \psi _{a'}^F| _{\mathcal {F}_h \backslash F} = 0. \end{aligned}$$

For any \(a\in A_{h,k+1}\), define the local spaces

$$\begin{aligned} \begin{aligned} \varSigma _{h,k+1,a}^{-1}&:= \mathrm {span} \{ \varphi _a^K T_{ij} ~|~ 1\le i \le j \le n, \bar{K} \ni a\}, \\ M_{h,k+1,a}&:= \mathrm{span}\{ \psi _a^F e_i ~|~ 1\le i \le n, F\in \mathcal {F}_h^i, \bar{F} \ni a\}, \end{aligned} \end{aligned}$$
(73)

where \(\{e_i~|~1 \le i \le n\}\) is the basis of \(\mathbb {R}^n\) and \(\{T_{ij}=\frac{1}{2}(e_ie_j^T + e_je_i^T)~|~1\le i \le j \le n\}\) is the basis of \(\mathbb {S}\). Clearly,

$$\begin{aligned} \varSigma _{h,k+1}^{-1} = \bigoplus _{a\in A_{h,k+1}} \varSigma _{h,k+1,a}^{-1} \quad \text {and} \quad M_{h,k+1} = \bigoplus _{a\in A_{h,k+1}} M_{h,k+1,a}. \end{aligned}$$

Moreover, if \(a\ne a'\) and \(\mu \in \mathcal {C}(\varSigma _{h,k+1,a}^{-1}) \cap \mathcal {C}(\varSigma _{h,k+1,a'}^{-1})\), then \(\mu \) vanishes at all the Lagrange nodes on the faces. This implies that \(\mu = 0\), namely

$$\begin{aligned} \mathcal {C}(\varSigma _{h,k+1,a}^{-1}) \cap \mathcal {C}(\varSigma _{h,k+1,a'}^{-1}) = \{0\} \qquad \text {if}~a \ne a'. \end{aligned}$$

Hence, we have

$$\begin{aligned} \mathrm {R}(\mathcal {C}) = \mathcal {C}(\varSigma _{h,k+1}^{-1}) = \bigoplus _{a\in A_{h,k+1}} \mathcal {C}(\varSigma _{h,k+1,a}^{-1}). \end{aligned}$$
(74)

Therefore, \(\mathrm {R}(\mathcal {C})\) has local basis since \(\mathcal {C}(\varSigma _{h,k+1,a}^{-1})\) is locally supported for any \(a \in A_{h,k+1}\).

Next, we construct the local basis for \(\mathrm {R}(\mathcal {C})^\perp \). Let

$$\begin{aligned} M_{h,k+1,a,{\perp }} := \left\{ \mu _a \in M_{h,k+1,a}~|~ \langle \mu _a, [\varvec{\tau }_a]\rangle _{\mathcal {F}^i_h} =0 \quad \forall \varvec{\tau }_a\in \varSigma _{h,k+1,a}^{-1} \right\} . \end{aligned}$$
(75)

If \(a \ne a'\), we further have

$$\begin{aligned} \langle \mu , \mathcal {C}\varvec{\tau }\rangle _{\mathcal {F}_h^i} = 0 \qquad \forall \mu \in M_{h,k+1,a}, \varvec{\tau }\in \varSigma _{h,k+1,a'}^{-1}. \end{aligned}$$

Hence, we have \(M_{h,k+1,a, {\perp }} \subset \mathrm {R}(\mathcal {C}) ^{\perp }\) and

$$\begin{aligned} \mathrm {R}(\mathcal {C}) ^{\perp } = \bigoplus _{a\in A_{h,k}} M_{h,k+1,a,{\perp }}. \end{aligned}$$
(76)

Therefore, the local basis \(\{\psi _1, \psi _2,\ldots ,\psi _{N_2}\}\) of \(\mathrm {R}(\mathcal {C}) ^{\perp }\) comes from the union of the basis of \(M_{h,k+1,a,{\perp }}\) for all \(a\in A_{h,k}\). \(\square \)

In the proof of Lemma  3, we show the decomposition of \(\mathrm {R}(\mathcal {C})^{\perp }\) in (76) as the direct sum of \(M_{h,k+1,a,{\perp }}\). Note that \(M_{h,k+1,a,{\perp }}\) only involves the local basis associated with the Lagrange node a. Moreover, the dimension of \(M_{h,k+1,a,\bot }\) only depends on the topology of the grid near the Lagrange node a, but does not depend on the degree of the polynomials k. Hence, we can use \(M_{h,k+1,a,{\perp }}\) to characterize singular vertices for any spatial dimension:

Definition 1

The Lagrange node a is called singular if \(M_{h,k+1,a,{\perp }}\) is a nontrivial set.

For 2D case, a direct calculation shows that \(M_{h,k+1,a,{\perp }}\) is a nontrivial set if and only if the edges meeting at this vertex a fall on two straight lines. For 3D case, when a is a Lagrange node on the edge e, \(M_{h,k+1,a,{\perp }}\) is a nontrivial set if and only if the faces meeting at this edge a fall on two planes. However, if a is a mesh vertex, it is difficult to describe the ordering of the element faces containing the vertex a. It is still an open problem to give a geometric description of the singular vertex in 3D case.

Appendix B: Proof of Lemma 6

Proof

The basis of \(M_{h,k+1,a,\bot }\) can be computed locally according to its definition (75). In particular, \(M_{h,k+1,a,\bot }\) is nontrivial for the 2D case only if a is an interior singular vertex. Thus, if there is no interior singular vertex in \(\mathcal {T}_h\), then \(\mathrm {R}(\mathcal {C})^\perp = \{0\}\), or \(M_{h,k+1} = \mathrm {R}(\mathcal {C}) = \bigoplus _{a\in A_{h,k+1}} \mathcal {C}(\varSigma _{h,k+1,a}^{-1})\). Further, a direct calculation shows that

$$\begin{aligned} \mathcal {C}(\varSigma _{h,k+1,a}^{-1}) = {\left\{ \begin{array}{ll} \mathrm {span}\{\varphi _a^Fe_i~|~ 1\le i \le 2, F\in \mathcal {F}_h^i, \bar{F} \ni a\} &{} a \in \bar{\mathcal {F}}_h^i, \\ \{0\} &{} a \not \in \bar{\mathcal {F}}_h^i, \end{array}\right. } \end{aligned}$$

where \(\varphi _a^F\) denotes the Lagrange nodal basis on F. Therefore, we can choose a special basis of \(\mathrm {R}(\mathcal {C})\) as

$$\begin{aligned} M_{h,k+1} = \bigoplus _{a \in \bar{F}_h^i \cap A_{h,k+1}} \mathrm {span}\{\varphi _a^F e_i ~|~ 1 \le i \le 2, F\in \mathcal {F}_h^i, \bar{F}\ni a\}. \end{aligned}$$
(77)

The mass matrix under the special basis (77) is the diagonal block matrix whose diagonal block entry is the local mass matrix under the Lagrange nodal basis on F. Hence, the mass matrix \(\varvec{M}\) is well-conditioned because the local mass matrix is well conditioned for the Lagrange nodal basis, which gives rise to (36). This completes the proof. \(\square \)

Appendix C: Proof of Lemma 7

Proof

In light of (74) in the proof of Lemma 6, there exists a Lagrange node \(a\in A_{h,k+1}\) such that \(\varphi _i \in \mathcal {C}(\varSigma _{h,k+1,a}^{-1})\). Further, we have

$$\begin{aligned} \varphi _i = \omega \varphi _a|_{\mathcal {F}_h}, \end{aligned}$$

where \(\varphi _a\) is the Lagrange nodal basis function at the node a and \(\omega \in L^2(\mathcal {F};{\mathbb {R}}^2)\) is piecewise constant and \(\mathrm {supp}(\omega ) \subset \{F\in \mathcal {F}_h^i~|~ \bar{F}\ni a\}\). Next, we construct \(\varvec{\tau }_i\in \varSigma _{h,k+1}^{-1}\) case by case according to the location of a. Clearly, if a is not located on the \(\bar{\mathcal {F}}_h^i\), then \(\mathcal {C}(\varSigma _{h,k+1,a}^{-1}) = \{0\}\). Hence, we only need to consider the following two cases: Internal Lagrange node on \(F\in \mathcal {F}_h^i\), or vertex of \(\mathcal {T}_h\). We first state a useful tool for the analysis: For any given vectors \(v, w \in \mathbb {R}^2\), there exists \(T\in {\mathbb {S}}\) such that

$$\begin{aligned} T v = w \quad \text {and} \quad \Vert T\Vert _{l^2} \le \sqrt{2} \frac{\Vert w\Vert _{l^2}}{\Vert v\Vert _{l^2}}. \end{aligned}$$
(78)

A straightforward calculation shows that T in (78) can be chosen as

$$\begin{aligned} T = \frac{w_1}{\Vert v\Vert _{l^2}^2} \begin{pmatrix} v_1 &{} v_2 \\ v_2 &{} -v_1 \end{pmatrix} + \frac{w_2}{\Vert v\Vert _{l^2}^2} \begin{pmatrix} -v_2 &{} v_1 \\ v_1 &{} v_2 \end{pmatrix}. \end{aligned}$$
Fig. 2
figure2

Internal Lagrange node on edge F

Case 1: Internal Lagrange node of\(F\in \mathcal {F}_h^i\). First, we select an element K such that \(F\in \bar{K}\) (cf. Fig. 2). By virtue of (78), there exists \(T\in {\mathbb {S}}\) such that

$$\begin{aligned} T \nu _F = \omega |_F \quad \text {and} \quad \Vert T\Vert _{l^2} \lesssim \Vert \omega |_F\Vert _{l^2}. \end{aligned}$$

From the definition of \(\varSigma _{h,k+1,a}^{-1}\) in (73), let \(\varvec{\tau }_i = \varphi _a^K T \in \varSigma _{h,k+1,a}^{-1}\). Then,

$$\begin{aligned} {[}\varvec{\tau }_i]_F= & {} \varphi _i|_F \quad \forall F\in \mathcal {F}_h \quad \text {and}\\&\quad \Vert \varvec{\tau }_i\Vert _{0}^2 = \Vert \varphi _a\Vert _{0, K}^2 \Vert T\Vert _{l^2}^2 \lesssim h \Vert \varphi _a\Vert _{0, F}^2 \Vert \omega |_F\Vert _{l^2}^2 = h\Vert \varphi _i\Vert _{0}^2. \end{aligned}$$
Fig. 3
figure3

Vertex of \(\mathcal {T}_h\). a Internal vertex. b Boundary vertex

Case 2: Vertex of\(\mathcal {T}_h\). Suppose that there are m (\(\ge 2\)) elements meeting at the vertex a. Since \(\kappa \ge \kappa _0>0\), there exist two adjacent elements (without loss of generality, denoted by \(K_1\) and \(K_2\)), such that the angles \(\theta _1\) and \(\theta _2\) satisfying \(|\theta _1+\theta _2-\pi | \ge \kappa _0\), (cf. Fig. 3). The edges that contain a are denoted by \(F_j\), \(1\le j \le m\) if a is an internal vertex, and \(1\le j \le m+1\) otherwise. If a is a boundary vertex, we further set \(F_1, F_{m+1} \in \mathcal {F}_h^\partial \), which is feasible because \(\kappa (a) \ge \kappa _0 >0\).

If a is an internal vertex, let \(F_{m+1}=F_1\) and \(\nu _{F_{m+1}} = \nu _{F_1}\). By virtue of (78), there exists \(T_m\in {\mathbb {S}}\) such that

$$\begin{aligned} T_m\nu _{F_{m+1}} = \omega |_{F_{m+1}} \quad \text {and}\quad \Vert T_m\Vert _{l^2} \lesssim \Vert \omega |_{F_{m+1}}\Vert _{l^2}. \end{aligned}$$
(79)

Note that \(T_m = \varvec{0}\in {\mathbb {S}}\) if a is a boundary vertex. Recursively for \(j=m-1,m-2,\ldots , 2\), there exist \(T_j \in {\mathbb {S}}\) on \(K_j\) such that

$$\begin{aligned} T_j \nu _{F_{j+1}}= & {} \omega |_{F_{j+1}} + T_{j+1} \nu _{F_{j+1}} \quad \text {and}\nonumber \\&\Vert T_j\Vert _{l^2} \lesssim \Vert \omega |_{F_{j+1}}\Vert _{l^2} + \Vert T_{j+1}\Vert _{l^2} \lesssim \sum _{s=j}^{m+1} \Vert \omega |_{F_{s}}\Vert _{l^2}. \end{aligned}$$
(80)

Since \(\omega |_{F_0} = 0\) if a is a boundary vertex, we simply set \(T_1 = \varvec{0}\in {\mathbb {S}}\).

Next, we find two symmetric matrices \(\tilde{T}_1 = c_1 t_1t_1^T\) and \(\tilde{T}_2 = c_2 t_3t_3^T\) on \(K_1\) and \(K_2\), respectively. Here, \(t_1, t_3\) are the unit tangential vectors of \(F_1\) and \(F_3\), respectively (cf. Fig. 3). The coefficients \(c_1,c_2\) are determined by

$$\begin{aligned} \tilde{T}_1\nu _{F_2} - \tilde{T}_2\nu _{F_2} = \omega |_{F_2} + T_2 \nu _{F_2}, \end{aligned}$$
(81)

i.e.

$$\begin{aligned} -\big ( t_1, t_3 \big )\begin{pmatrix} c_1\sin \theta _1\\ c_2\sin \theta _2 \end{pmatrix} = \omega |_{F_2} + T_2 \nu _{F_2}. \end{aligned}$$

Since \(|\theta _1+\theta _2-\pi | \ge \kappa _0\), we have \(|\det (t_1,t_3)| = |t_1\times t_3| = |\sin (\theta _1 +\theta _2) | \ge \sin (\kappa _0)\). Thus, the matrix \((t_1, t_3)\) is invertible. Moreover, we have \(| (t_1,t_3)^{-1} |_{\infty } \lesssim \sin ^{-1}(\kappa _0) \) and, by the shape regularity of grids, \(|\sin \theta _1|\) and \(|\sin \theta _2|\) are bounded uniformly away from zero. Thus,

$$\begin{aligned} \Vert \tilde{T}_1\Vert _{l^2}^2+\Vert \tilde{T}_1\Vert _{l^2}^2 \lesssim c_1^2+c_2^2 \lesssim \sin ^{-2}(\kappa _0) \Vert \omega |_{F_2} + T_2 \nu _{F_2} \Vert _{l^2}^2 \lesssim \sin ^{-2}(\kappa _0) \sum _{j=1}^{m+1}\Vert \omega |_{F_j}\Vert _{l^2}^2. \end{aligned}$$

In light of (79), (80), and (81), let

$$\begin{aligned} \varvec{\tau }_i |_{K_j} = {\left\{ \begin{array}{ll} \varphi _a^{K_j}(T_j + \tilde{T}_j) &{} j=1,2,\\ \varphi _a^{K_j} T_j &{} 3\le j \le m. \end{array}\right. } \end{aligned}$$
(82)

Then, we have

$$\begin{aligned} {[}\varvec{\tau }_i]|_{F} = \varphi _i|_F \quad \forall F\in \mathcal {F}_h,\quad \text {and} \quad \Vert \varvec{\tau }_i\Vert _0^2 \lesssim h \sin ^{-2}(\kappa _0) \Vert \varphi _i\Vert _{0}^2. \end{aligned}$$

This completes the proof. \(\square \)

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Gong, S., Wu, S. & Xu, J. New hybridized mixed methods for linear elasticity and optimal multilevel solvers. Numer. Math. 141, 569–604 (2019). https://doi.org/10.1007/s00211-018-1001-3

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Mathematics Subject Classification

  • 65N30
  • 65N55