## On the implementation of mixed methods as nonconforming methods for second-order elliptic problems

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- by Todd Arbogast and Zhangxin Chen PDF
- Math. Comp.
**64**(1995), 943-972 Request permission

## Abstract:

In this paper we show that mixed finite element methods for a fairly general second-order elliptic problem with variable coefficients can be given a nonmixed formulation. (Lower-order terms are treated, so our results apply also to parabolic equations.) We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection method’s finite element space ${M_h}$ satisfies three conditions, then the two approximation methods are equivalent. These three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. We then construct appropriate nonconforming spaces ${M_h}$ for the known triangular and rectangular elements. The lowest-order Raviart-Thomas mixed solution on rectangular finite elements in ${\mathbb {R}^2}$ and ${\mathbb {R}^3}$, on simplices, or on prisms, is then implemented as a nonconforming method modified in a simple and computationally trivial manner. This new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution. A rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal-order multigrid methods for the solution of the linear system.## References

- D. N. Arnold and F. Brezzi,
*Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates*, RAIRO Modél. Math. Anal. Numér.**19**(1985), no. 1, 7–32 (English, with French summary). MR**813687**, DOI 10.1051/m2an/1985190100071 - Randolph E. Bank and Todd Dupont,
*An optimal order process for solving finite element equations*, Math. Comp.**36**(1981), no. 153, 35–51. MR**595040**, DOI 10.1090/S0025-5718-1981-0595040-2 - Susanne C. Brenner,
*An optimal-order multigrid method for $\textrm {P}1$ nonconforming finite elements*, Math. Comp.**52**(1989), no. 185, 1–15. MR**946598**, DOI 10.1090/S0025-5718-1989-0946598-X - Susanne C. Brenner,
*A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method*, SIAM J. Numer. Anal.**29**(1992), no. 3, 647–678. MR**1163350**, DOI 10.1137/0729042 - F. Brezzi,
*On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge**8**(1974), no. R-2, 129–151 (English, with French summary). MR**365287** - Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin,
*Mixed finite elements for second order elliptic problems in three variables*, Numer. Math.**51**(1987), no. 2, 237–250. MR**890035**, DOI 10.1007/BF01396752 - Franco Brezzi, Jim Douglas Jr., Michel Fortin, and L. Donatella Marini,
*Efficient rectangular mixed finite elements in two and three space variables*, RAIRO Modél. Math. Anal. Numér.**21**(1987), no. 4, 581–604 (English, with French summary). MR**921828**, DOI 10.1051/m2an/1987210405811 - Franco Brezzi, Jim Douglas Jr., and L. D. Marini,
*Two families of mixed finite elements for second order elliptic problems*, Numer. Math.**47**(1985), no. 2, 217–235. MR**799685**, DOI 10.1007/BF01389710
Zhangxin Chen, - Zhangxin Chen,
*Analysis of mixed methods using conforming and nonconforming finite element methods*, RAIRO Modél. Math. Anal. Numér.**27**(1993), no. 1, 9–34 (English, with English and French summaries). MR**1204626**, DOI 10.1051/m2an/1993270100091 - Zhangxin Chen and Bernardo Cockburn,
*Analysis of a finite element method for the drift-diffusion semiconductor device equations: the multidimensional case*, Numer. Math.**71**(1995), no. 1, 1–28. MR**1339730**, DOI 10.1007/s002110050134 - Z. Chen and J. Douglas Jr.,
*Prismatic mixed finite elements for second order elliptic problems*, Calcolo**26**(1989), no. 2-4, 135–148 (1990). MR**1083050**, DOI 10.1007/BF02575725 - Zhangxin Chen and Jim Douglas Jr.,
*Approximation of coefficients in hybrid and mixed methods for nonlinear parabolic problems*, Mat. Apl. Comput.**10**(1991), no. 2, 137–160 (English, with Portuguese summary). MR**1172090** - Jim Douglas Jr. and Jean E. Roberts,
*Global estimates for mixed methods for second order elliptic equations*, Math. Comp.**44**(1985), no. 169, 39–52. MR**771029**, DOI 10.1090/S0025-5718-1985-0771029-9 - Lucia Gastaldi and Ricardo Nochetto,
*Optimal $L^\infty$-error estimates for nonconforming and mixed finite element methods of lowest order*, Numer. Math.**50**(1987), no. 5, 587–611. MR**880337**, DOI 10.1007/BF01408578 - J. Mandel, S. McCormick, and R. Bank,
*Variational multigrid theory*, Multigrid methods, Frontiers Appl. Math., vol. 3, SIAM, Philadelphia, PA, 1987, pp. 131–177. MR**972757** - Luisa Donatella Marini,
*An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method*, SIAM J. Numer. Anal.**22**(1985), no. 3, 493–496. MR**787572**, DOI 10.1137/0722029 - J.-C. Nédélec,
*Mixed finite elements in $\textbf {R}^{3}$*, Numer. Math.**35**(1980), no. 3, 315–341. MR**592160**, DOI 10.1007/BF01396415 - J.-C. Nédélec,
*A new family of mixed finite elements in $\textbf {R}^3$*, Numer. Math.**50**(1986), no. 1, 57–81. MR**864305**, DOI 10.1007/BF01389668 - P.-A. Raviart and J. M. Thomas,
*A mixed finite element method for 2nd order elliptic problems*, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR**0483555** - Rolf Stenberg,
*A family of mixed finite elements for the elasticity problem*, Numer. Math.**53**(1988), no. 5, 513–538. MR**954768**, DOI 10.1007/BF01397550

*On the relationship between mixed and Galerkin finite element methods*, Ph. D. Thesis, Purdue University, West Lafayette, Indiana, 1991. —,

*Unified analysis of the hybrid form of mixed finite elements for second order elliptic problems*, J. Engrg. Math.

**8**(1991), 91-102.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp.
**64**(1995), 943-972 - MSC: Primary 65N30; Secondary 65N22
- DOI: https://doi.org/10.1090/S0025-5718-1995-1303084-8
- MathSciNet review: 1303084