# Stability of Lagrange elements for the mixed Laplacian

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## Abstract

The stability properties of simple element choices for the mixed formulation of the Laplacian are investigated numerically. The element choices studied use vector Lagrange elements, i.e., the space of continuous piecewise polynomial vector fields of degree at most *r*, for the vector variable, and the divergence of this space, which consists of discontinuous piecewise polynomials of one degree lower, for the scalar variable. For polynomial degrees *r* equal 2 or 3, this pair of spaces was found to be stable for all mesh families tested. In particular, it is stable on diagonal mesh families, in contrast to its behavior for the Stokes equations. For degree *r* equal 1, stability holds for some meshes, but not for others. Additionally, convergence was observed precisely for the methods that were observed to be stable. However, it seems that optimal order *L* ^{2} estimates for the vector variable, known to hold for *r*>3, do not hold for lower degrees.

## Keywords

Mixed finite elements Lagrange finite elements Stability## Mathematics Subject Classification (2000)

65N30## Preview

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## References

- 1.Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer.
**15**, 1–155 (2006) CrossRefMathSciNetzbMATHGoogle Scholar - 2.Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math.
**20**, 179–192 (1972/73) CrossRefMathSciNetGoogle Scholar - 3.Boffi, D., Brezzi, F., Fortin, M.: Finite elements for the Stokes problem. In: Boffi, D., Gastaldi, L. (eds.) Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics, vol. 1939, pp. 45–100. Springer, Berlin (2008). Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1, 2006 CrossRefGoogle Scholar
- 4.Boffi, D., Brezzi, F., Gastaldi, L.: On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput.
**69**(229), 121–140 (2000) MathSciNetzbMATHGoogle Scholar - 5.Boffi, D., Duran, R.G., Gastaldi, L.: A remark on spurious eigenvalues in a square. Appl. Math. Lett.
**12**(3), 107–114 (1999) CrossRefMathSciNetzbMATHGoogle Scholar - 6.Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Texts in Applied Mathematics, vol. 15. Springer, New York (2008) zbMATHGoogle Scholar
- 7.Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Fr. Autom. Inf. Rech. Opér. Sér. Rouge
**8**(R-2), 129–151 (1974) MathSciNetGoogle Scholar - 8.Brezzi, F., Douglas, J. Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math.
**47**(2), 217–235 (1985) CrossRefMathSciNetzbMATHGoogle Scholar - 9.Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991) zbMATHGoogle Scholar
- 10.Chapelle, D., Bathe, K.-J.: The inf-sup test. Comput. Struct.
**47**(4–5), 537–545 (1993) CrossRefMathSciNetzbMATHGoogle Scholar - 11.Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw.
**31**(3), 351–362 (2005) CrossRefMathSciNetzbMATHGoogle Scholar - 12.Logg, A., Wells, G.N. et al.: DOLFIN. http//www.fenics.org/dolfin/
- 13.Malkus, D.S.: Eigenproblems associated with the discrete LBB condition for incompressible finite elements. Int. J. Eng. Sci.
**19**(10), 1299–1310 (1981) CrossRefMathSciNetzbMATHGoogle Scholar - 14.Mardal, K.A., Tai, X.-C., Winther, R.: A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal.
**40**(5), 1605–1631 (2002) (electronic) CrossRefMathSciNetzbMATHGoogle Scholar - 15.Morgan, J., Scott, L.R.: The dimension of the space of
*C*^{1}piecewise polynomials. Research Report UH/MD 78, University of Houston, Mathematics Department, 1990 Google Scholar - 16.Morgan, J., Scott, R.: A nodal basis for
*C*^{1}piecewise polynomials of degree*n*≥5. Math. Comput.**29**, 736–740 (1975) CrossRefMathSciNetzbMATHGoogle Scholar - 17.Qin, J.: On the convergence of some low order mixed finite elements for incompressible fluids. PhD thesis, Penn State (1994) Google Scholar
- 18.Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods. Proc. Conf. Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975. Lecture Notes in Math., vol. 606, pp. 292–315. Springer, Berlin (1977) CrossRefGoogle Scholar
- 19.Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér.
**19**(1), 111–143 (1985) MathSciNetzbMATHGoogle Scholar