A two-level enriched finite element method for a mixed problem

Authors:
Alejandro Allendes, Gabriel R. Barrenechea, Erwin Hernández and Frédéric Valentin

Journal:
Math. Comp. **80** (2011), 11-41

MSC (2010):
Primary 65N30, 65N12; Secondary 76S99

DOI:
https://doi.org/10.1090/S0025-5718-2010-02364-6

Published electronically:
July 26, 2010

MathSciNet review:
2728970

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Abstract | References | Similar Articles | Additional Information

Abstract: The simplest pair of spaces is made inf-sup stable for the mixed form of the Darcy equation. The key ingredient is to enhance the finite element spaces inside a Petrov-Galerkin framework with functions satisfying element-wise local Darcy problems with right hand sides depending on the residuals over elements and edges. The enriched method is symmetric, locally mass conservative and keeps the degrees of freedom of the original interpolation spaces. First, we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms. Then, a low cost two-level finite element method is proposed to effectively obtain enhancing basis functions. The approach lays on a two-scale numerical analysis and shows that well-posedness and optimality is kept, despite the second level numerical approximation. Several numerical experiments validate the theoretical results and compares (favourably in some cases) our results with the classical Raviart-Thomas element.

**1.**R. Araya, G. R. Barrenechea, L. P. Franca, and F. Valentin,*Stabilization arising from PGEM: A review and further developments*, Applied Numerical Mathematics**59**(2009), no. 9, 2065-2081. MR**2532854****2.**R. Araya, G. R. Barrenechea, and F. Valentin,*Stabilized finite element methods based on multiscale enrichment for the Stokes problem*, SIAM J. Numer. Anal.**44**(2006), no. 1, 322-348. MR**2217385 (2007f:65049)****3.**T. Arbogast,*Analysis of a two-scale locally conservative subgrid upscaling for elliptic problems*, SIAM J. Numer. Anal.**42**(2004), no. 2, 576-598. MR**2084227 (2005h:65205)****4.**C. Baiocchi, F. Brezzi, and L. P. Franca,*Virtual bubbles and Galerkin-Least-Squares type methods (Ga.L.S.)*, Comput. Methods Appl. Mech. Engrg.**105**(1993), no. 1, 125-141. MR**94g:65058****5.**G. R. Barrenechea, L. P. Franca, and F. Valentin,*A Petrov-Galerkin enriched method: A mass conservative finite element method for the Darcy equation*, Computer Methods in Applied Mechanics and Engineering**196**(2007), no. 21-24, 2449-2464. MR**2319050 (2008e:76108)****6.**-,*A symmetric nodal conservative finite element method for the Darcy equation*, SIAM J. Numer. Anal.**47**(2009), no. 5, 3652-3677. MR**2576515****7.**G. R. Barrenechea and F. Valentin,*An unusual stabilized finite element method for a generalized Stokes problem*, Numer. Math.**92**(2002), no. 4, 653-677. MR**2003i:76063****8.**-,*Relationship between multiscale enrichment and stabilized finite element methods for the generalized Stokes problem*, CRAS**341**(2005), no. 10, 635-640. MR**2179805****9.**P. B. Bochev and M. D. Gunzburger,*A locally conservative least-squares method for Darcy flows*, Comm. Numer. Methods Engrg.**24**(2008), no. 2, 97-110. MR**2369638****10.**F. Brezzi, B. Cockburn, L. D. Marini, and E. Süli,*Stabilization mechanisms in discontinuous Galerkin finite element methods*, Comput. Methods Appl. Mech. Engrg.**195**(2006), no. 25-28, 3293-3310. MR**2220920 (2006m:65256)****11.**F. Brezzi and M. Fortin,*Mixed and hybrid finite element methods*, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, Berlin, New-York, 1991. MR**1115205 (92d:65187)****12.**F. Brezzi, L. P. Franca, and A. Russo,*Further considerations on residual-free bubbles for advective-diffusive equations*, Comput. Methods Appl. Mech. Engrg.**166**(1998), no. 1-2, 25-33. MR**99j:65197****13.**F. Brezzi, L. P. Franca, T. J. R. Hughes, and A. Russo,*Stabilization techniques and subgrid scale capturing*, Proceedings of the Conference ``The State of the Art in Numerical Analysis'', York, 1-4 April, 1996 (I. Duff, ed.), IMA Conference, Oxford University Press, 1997. MR**1628354 (99f:65162)****14.**F. Brezzi and A. Russo,*Choosing bubbles for advection-diffusion problems*, Math. Models Methods Appl. Sci.**4**(1994), no. 4, 571-587. MR**1291139 (95h:76079)****15.**E. Burman and P. Hansbo,*A unified stabilized method for Stokes' and Darcy's equations*, J. Comput. Appl. Math.**198**(2007), no. 1, 35-51. MR**2250387 (2007i:65076)****16.**Z. Chen and T. Hou,*A mixed multiscale finite element method for elliptic problems with oscillating coefficients*, Mathematics of Computation**72**(2003), no. 242, 541-576. MR**1954956 (2004a:65147)****17.**P. Clément,*Approximation by finite element functions using local regularization*, RAIRO Anal. Numér. (1975), no. 9, 77-84. MR**0400739 (53:4569)****18.**W. E and B. Engquist,*The heterogeneous multiscale methods*, Commun. Math. Sci.**1**(2003), no. 1, 87-132. MR**2004b:35019****19.**A. Ern and J.-L. Guermond,*Theory and practice of finite elements*, Springer-Verlag, 2004. MR**2050138 (2005d:65002)****20.**L. P. Franca, A. L. Madureira, L. Tobiska, and F. Valentin,*Convergence analysis of a multiscale finite element method for singularly perturbed problems*, SIAM Multiscale Model. and Simul.**4**(2005), no. 3, 839-866. MR**2203943 (2006k:65316)****21.**L. P. Franca, A. L. Madureira, and F. Valentin,*Towards multiscale functions: Enriching finite element spaces with local but not bubble-like functions*, Comput. Methods Appl. Mech. Engrg.**194**(2005), 3006-3021. MR**2142535 (2006a:65159)****22.**L. P. Franca, A. Nesliturk, and M. Stynes,*On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method*, Comput. Methods Appl. Mech. Engrg.**166**(1998), 35-49. MR**1660133 (99k:65107)****23.**V. Girault and P. A. Raviart,*Finite element methods for Navier-Stokes equations: Theory and algorithms*, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, New-York, 1986. MR**851383 (88b:65129)****24.**T. J. R. Hughes, G. R. Feijoo, L. Mazzei, and J. Quincy,*The variational multiscale method -- a paradigm for computational mechanics*, Computer Methods in Applied Mechanics and Engineering**166**(1998), no. 1-2, 3-24. MR**1660141 (99m:65239)****25.**L. E. Payne and H. F. Weinberger,*An optimal Poincaré inequality for convex domains*, Arch. Rational Mech. Anal.**5**(1960), 286-292. MR**0117419 (22:8198)****26.**P. A. Raviart and J. M. Thomas,*A mixed finite element method for 2nd order elliptic problems*, Mathematical aspect of finite element methods, no. 606 in Lecture Notes in Mathematics, pp. 292-315, Springer-Verlag, New York, 1977. MR**0483555 (58:3547)****27.**A. Russo,*Residual free bubbles and stabilized methods*, Proceedings of the Ninth International Conference on Finite Elements in Fluids - New Trends and Applications (Venice, Italy) (M. Morandi Cecchi, K. Morgan, J. Periaux, B. A. Schrefler, and O. C. Zienkiewicz, eds.), October 1995, pp. 1607-1615.

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Additional Information

**Alejandro Allendes**

Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom

Email:
alejandro.allendes-flores@strath.ac.uk

**Gabriel R. Barrenechea**

Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom

Email:
gabriel.barrenechea@strath.ac.uk

**Erwin Hernández**

Affiliation:
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile

Email:
erwin.hernandez@usm.cl

**Frédéric Valentin**

Affiliation:
Departamento de Matemática Aplicada e Computacional, Laboratório Nacional de Computação Científica, Av. Getúlio Vargas, 333, 25651-070 Petrópolis - RJ, Brazil

Email:
valentin@lncc.br

DOI:
https://doi.org/10.1090/S0025-5718-2010-02364-6

Keywords:
Darcy flow,
enriched finite element method,
Petrov-Galerkin approach,
mass conservation,
two-level finite element method

Received by editor(s):
October 10, 2008

Received by editor(s) in revised form:
July 1, 2009

Published electronically:
July 26, 2010

Additional Notes:
The second author was partially supported by Starter’s Grant, Faculty of Sciences, University of Strathclyde.

The third author was supported by CONICYT Chile, through FONDECYT Project No. 1070276 and by Universidad Santa María through project No. DGIP-USM 120851.

The fourth author was supported by CNPq /Brazil Grant No. 304051/2006-3, FAPERJ/Brazil Grant No. E-26/100.519/2007.

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.