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Introductory Material and Finite Element Methods

  • Alexandre L. Madureira
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In this chapter, we introduce some notation, and also state some basic results regarding the Galerkin Method. In particular, some elementary estimates are presented, highlighting the importance of coercivity constants. This chapter also contains a brief introduction to some alternative methods, such as the Residual Free Bubble Method, the Multiscale Finite Element Method, the Localized Orthogonal Decomposition, the Variational Multiscale Method, and Hybrid and Stabilized Methods.

Keywords

Finite Element Method Hybridizable Discontinuous Galerkin Galerkin Solution Continuous Piecewise Linear Function Multiscale Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 15.
    Araya, R., Harder, C., Paredes, D., & Valentin, F. (2013). Multiscale hybrid-mixed method. SIAM Journal of Numerical Analysis, 51(6), 3505–3531. doi:10.1137/120888223. MR3143841.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 16.
    Arbogast, T., Pencheva, G., Wheeler, M. F., & Yotov, I. (2007). A multiscale mortar mixed finite element method. Multiscale Modelling and Simulation, 6(1), 319–346. doi:10.1137/060662587. MR2306414.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 17.
    Arnold, D. N., & Brezzi, F. (1985). Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. RAIRO Modélisation Mathématique et Analyse Numérique, 19(1), 7–32 (English, with French summary). MR813687Google Scholar
  4. 18.
    Arnold, D. N., Brezzi, F., Cockburn, B., & Marini, L. D. (2001/2002). Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal of Numerical Analysis, 39(5), 1749–1779. doi:10.1137/S0036142901384162. MR1885715 (2002k:65183).Google Scholar
  5. 20.
    Atkinson, K., & Han, W. (2005). Theoretical numerical analysis. A functional analysis framework. Texts in Applied Mathematics (2nd ed., Vol. 39). New York: Springer. MR2153422 (2006a:65001).Google Scholar
  6. 22.
    Axelsson, O., & Barker, V. A. (2001). Finite element solution of boundary value problems: Theory and computation. Classics in Applied Mathematics (Vol. 35). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); Reprint of the 1984 original. MR1856818 (2002g:65001)Google Scholar
  7. 23.
    Babuška, I. (1970/1971). Error-bounds for finite element method. Numerical Mathematics, 16, 322–333. MR0288971 (44#6166).Google Scholar
  8. 31.
    Boffi, D., Brezzi, F., & Fortin, M. (2013). Mixed finite element methods and applications. Springer Series in Computational Mathematics (Vol. 44). Heidelberg: Springer. MR3097958.Google Scholar
  9. 34.
    Brenner, S. C., & Scott, L. R. (2008). The mathematical theory of finite element methods. Texts in Applied Mathematics (3rd ed., Vol. 15). New York: Springer. MR2373954 (2008m:65001).Google Scholar
  10. 38.
    Brezis, H. (2011). Functional analysis, Sobolev spaces and partial differential equations. New York: Universitext, Springer. MR2759829.Google Scholar
  11. 39.
    Brezzi, F., Bristeau, M. O., Franca, L. P., Mallet, M., & Rogé, G. (1992). A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Computational Methods in Applied Mechanical Engineering, 96(1), 117–129. doi:10.1016/0045-7825(92)90102-P. MR1159592 (92k:76056).Google Scholar
  12. 40.
    Brezzi, F., Franca, L. P., Hughes, T. J. R., & Russo, A. (1997). \(b =\int g\), Computational Methods in Applied Mechanical Engineering, 145(3–4), 329–339. doi:10.1016/S0045-7825(96)01221-2. MR1456019 (98g:65086).Google Scholar
  13. 43.
    Brezzi, F., & Russo, A. (1994). Choosing bubbles for advection-diffusion problems. Mathematical Models and Methods of Applied Science, 4(4), 571–587. MR1291139 (95h:76079).Google Scholar
  14. 44.
    Brooks, A. N., & Hughes, T. J. R. (1982). Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computational Methods in Applied Mechanical Engineering, 32(1–3), 199–259. FENOMECH ’81, Part I (Stuttgart, 1981). doi:10.1016/0045-7825(82)90071-8. MR679322 (83k:76005).Google Scholar
  15. 54.
    Ciarlet, P. G. (2002). The finite element method for elliptic problems. Classics in Applied Mathematics (Vol. 40). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR1930132.Google Scholar
  16. 55.
    Ciarlet, P. G. (2013). Linear and nonlinear functional analysis with applications. Philadelphia, PA: Society for Industrial and Applied Mathematics. MR3136903.Google Scholar
  17. 58.
    Cockburn, B., Dong, B., Guzmán, J., Restelli, M., & Sacco, R. (2009). A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM Journal of Scientific Computing, 31(5), 3827–3846. doi:10.1137/080728810. MR2556564 (2010m:65216).Google Scholar
  18. 59.
    Cockburn, B., Gopalakrishnan, J., & Lazarov, R. (2009). Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM Journal of Numerical Analysis, 47(2), 1319–1365. doi:10.1137/070706616. MR2485455.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 61.
    Coutinho, A. L. G. A., Franca, L. P., & Valentin, F. (2012). Numerical multiscale methods. International Journal for Numerical Methods in Fluids, 70(4), 403–419. doi:10.1002/fld.2727. MR2974524.MathSciNetCrossRefGoogle Scholar
  20. 65.
    Dautray, R., & Lions, J.-L. (1988). Mathematical analysis and numerical methods for science and technology (Vol. 2). Berlin: Springer. Functional and variational methods; With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Héléne Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily; Translated from the French by Ian N. Sneddon. MR969367 (89m:00001).Google Scholar
  21. 69.
    Di Pietro, D. A., & Ern, A. (2015). A hybrid high-order locking-free method for linear elasticity on general meshes. Computational Methods in Applied Mechanical Engineering, 283, 1–21. doi:10.1016/j.cma.2014.09.009. MR3283758.MathSciNetCrossRefGoogle Scholar
  22. 71.
    Dostàl, Z., Horàk, D., & Kučera, R. (2006). Total FETI—an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Communications in Numerical Methods in Engineering, 22(12), 1155–1162. doi:10.1002/cnm.881. MR2282408 (2007k:65177).Google Scholar
  23. 73.
    Efendiev, Y., Galvis, J., & Hou, T. Y. (2013). Generalized multiscale finite element methods (GMsFEM). Journal of Computational Physics, 251, 116–135. doi:10.1016/ j.jcp.2013.04.045. MR3094911.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 75.
    Efendiev, Y., & Hou, T. Y. (2008). Multiscale computations for flow and transport in heterogeneous media. In Quantum transport. Lecture Notes in Mathematics (Vol. 1946, pp. 169–248). Berlin: Springer. MR2497877.Google Scholar
  25. 76.
    Efendiev, Y., & Hou, T. Y. (2009). Multiscale finite element methods: Theory and applications. Surveys and Tutorials in the Applied Mathematical Sciences (Vol. 4). New York: Springer. MR2477579.Google Scholar
  26. 79.
    Efendiev, Y., Lazarov, R., Moon, M., & Shi, K. (2015). A spectral multiscale hybridizable discontinuous Galerkin method for second order elliptic problems. Computational Methods in Applied Mechanical Engineering, 292, 243–256. doi:10.1016/j.cma.2014.09.036. MR3347248.MathSciNetCrossRefGoogle Scholar
  27. 80.
    Efendiev, Y., & Pankov, A. (2003). Numerical homogenization of monotone elliptic operators. Multiscale Modelling and Simulation, 2(1), 62–79 (electronic). MR2044957 (2005a:65153).Google Scholar
  28. 81.
    Efendiev, Y. R., & Wu, X.-H. (2002). Multiscale finite element for problems with highly oscillatory coefficients. Numerical Mathematics, 90(3), 459–486. MR1884226 (2002m:65114).Google Scholar
  29. 82.
    Elfverson, D., Georgoulis, E. H., Målqvist, A., & Peterseim, D. (2013). Convergence of a discontinuous Galerkin multiscale method. SIAM Journal of Numerical Analysis, 51(6), 3351–3372. doi:10.1137/120900113. MR3141754.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 85.
    Ern, A., & Guermond, J.-L. (2004). Theory and practice of finite elements. Applied Mathematical Sciences (Vol. 159). New York: Springer. MR2050138 (2005d:65002).Google Scholar
  31. 87.
    Farhat, C., Harari, I., & Franca, L. P. (2001). The discontinuous enrichment method. Computational Methods in Applied Mechanical Engineering, 190(48), 6455–6479. doi:10.1016/S0045-7825(01)00232-8. MR1870426 (2002j:76083).Google Scholar
  32. 88.
    Farhat, C., & Roux, F.-X. (1991). A method of finite element tearing and interconnecting and its parallel solution algorithm. International Journal for Numerical Methods in Engineering, 32(6), 1205–1227. doi:10.1002/nme.1620320604.CrossRefzbMATHGoogle Scholar
  33. 89.
    Franca, L. P. & Dutra do Carmo, E. G. (1989). The Galerkin gradient least-squares method. Computational Methods in Applied Mechanical Engineering, 74(1), 41–54. doi:10.1016/0045-7825(89)90085-6. MR1017749 (90i:65195).Google Scholar
  34. 90.
    Franca, L. P., Farhat, C., Macedo, A. P., & Lesoinne, M. (1997). Residual-free bubbles for the Helmholtz equation. International Journal for Numerical Methods in Engineering, 40(21), 4003–4009. MR1475348.Google Scholar
  35. 91.
    Franca, L. P., Frey, S. L., & Hughes, T. J. R. (1992). Stabilized finite element methods, I: Application to the advective-diffusive model. Computational Methods in Applied Mechanical Engineering, 95(2), 253–276. doi:10.1016/0045-7825(92)90143-8. MR1155924 (92m:76089).Google Scholar
  36. 92.
    Franca, L. P., & Hwang, F.-N. (2002). Refining the submesh strategy in the two-level finite element method: Application to the advection-diffusion equation. International Journal for Numerical Methods in Fluids, 39(2), 161–187. doi:10.1002/fld.219. MR1903572.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 93.
    Franca, L. P., & Madureira, A. L. (1993). Element diameter free stability parameters for stabilized methods applied to fluids. Computational Methods in Applied Mechanical Engineering, 105(3), 395–403. doi:10.1016/0045-7825(93)90065-6. MR1224304 (94g:76033).Google Scholar
  38. 95.
    Franca, L. P., Madureira, A. L., & Valentin, F. (2005).Towards multiscale functions: Enriching finite element spaces with local but not bubble-like functions. Computational Methods in Applied Mechanical Engineering, 194(27–29), 3006–3021. MR2142535 (2006a:65159).Google Scholar
  39. 96.
    Franca, L. P., Nesliturk, A., & Stynes, M. (1998). On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method. Computational Methods in Applied Mechanical Engineering, 166(1–2), 35–49. doi:10.1016/S00457825(98)00081-4. MR1660133.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 99.
    Franca, L. P., & Russo, A. (1996). Approximation of the Stokes problem by residual-free macro bubbles. East-West Journal of Numerical Mathematics, 4(4), 265–278. MR1430240 (97i:76076).Google Scholar
  41. 100.
    Franca, L. P., & Russo, A. (1996). Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles. Applied Mathematics Letters, 9(5), 83–88. MR1415477 (97e:65121).Google Scholar
  42. 102.
    Franca, L. P., & Russo, A. (1997). Mass lumping emanating from residual-free bubbles. Computational Methods in Applied Mechanical Engineering, 142(3–4), 353–360. MR1442384 (98c:76064).Google Scholar
  43. 103.
    Galeão, A. C., Almeida, R. C., Malta, S. M. C., & Loula, A. F. D. (2004). Finite element analysis of convection dominated reaction-diffusion problems. Applied Numerical Mathematics, 48(2), 205–222. doi:10.1016/j.apnum.2003.10.002. MR2029331 (2004k:65219).Google Scholar
  44. 104.
    Gatica, G. N. (2014). A simple introduction to the mixed finite element method: Theory and applications. Springer Briefs in Mathematics. Cham: Springer. MR3157367.Google Scholar
  45. 106.
    Girault, V., & Raviart, P.-A. (1986). Finite element methods for Navier-Stokes equations: Theory and algorithms. Springer Series in Computational Mathematics (Vol. 5). Berlin: Springer. MR851383 (88b:65129).Google Scholar
  46. 107.
    Glowinski, R., & Wheeler, M. F. (1988). Domain decomposition and mixed finite element methods for elliptic problems. In Partial differential equations (Paris, 1987) (pp. 144–172). Philadelphia, PA: SIAM. MR972516 (90a:65237).Google Scholar
  47. 108.
    Golub, G. H., & Van Loan, C. F. (1983). Matrix computations. Johns Hopkins Series in the Mathematical Sciences (Vol. 3). Baltimore, MD: Johns Hopkins University Press. MR733103 (85h:65063).Google Scholar
  48. 109.
    Grisvard, P. (2011). Elliptic problems in nonsmooth domains. Classics in Applied Mathematics (Vol. 69). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Reprint of the 1985 original [MR0775683]; With a foreword by Susanne C. Brenner. MR3396210.Google Scholar
  49. 110.
    Guzmán, J. (2006). Local analysis of discontinuous Galerkin methods applied to singularly perturbed problems. Journal of Numerical Mathematics, 14(1), 41–56. doi:10.1163/156939506776382157. MR2229818 (2007b:65122).Google Scholar
  50. 112.
    Hackbusch, W. (2010). Elliptic differential equations: Theory and numerical treatment. Reprint of the 1992 English edition. Springer Series in Computational Mathematics (Vol. 18). Berlin: Springer. Translated from the 1986 corrected German edition by Regine Fadiman and Patrick D. F. Ion. MR2683186.Google Scholar
  51. 114.
    Harari, I., & Hughes, T. J. R. (1994). Stabilized finite element methods for steady advection-diffusion with production. Computational Methods in Applied Mechanical Engineering, 115(1–2), 165–191. doi:10.1016/0045-7825(94)90193-7. MR1278815 (95a:76059).Google Scholar
  52. 115.
    Harder, C., Madureira, A.L., & Valentin, F. (2016). A hybrid-mixed method for elasticity. ESAIM Mathematical Modelling and Numerical Analysis, 50(2), 311–336. ISSN: 0764-583X; doi:10.1051/m2an/2015046. MR3482545.Google Scholar
  53. 117.
    Harder, C., Paredes, D., & Valentin, F. (2015). On a multiscale hybrid-mixed method for advective-reactive dominated problems with heterogeneous coefficients. Multiscale Modelling and Simulation, 13(2), 491–518. doi:10.1137/130938499. MR3336297.MathSciNetCrossRefzbMATHGoogle Scholar
  54. 119.
    Henao, C. A. A., Coutinho, A. L. G. A., & Franca, L. P. (2010). A stabilized method for transient transport equations. Computational Mechanics, 46(1), 199–204. doi:10.1007/s00466-010-0465-5. MR2644409.MathSciNetCrossRefzbMATHGoogle Scholar
  55. 127.
    Hou, T. Y. (2003). Numerical approximations to multiscale solutions in partial differential equations. In Frontiers in numerical analysis (Durham, 2002) (pp. 241–301). MR2006969 (2004m:65219).Google Scholar
  56. 128.
    Hou, T. Y., & Liu, P. (2016). Optimal local multi-scale basis functions for linear elliptic equations with rough coefficient. Discrete and Continuous Dynamical Systems, 36(8), 4451–4476. ISSN:1078-0947; doi:10.3934/dcds.2016.36.4451. MR3479521.Google Scholar
  57. 132.
    Hughes, T. J. R. (1978). A simple scheme for developing ‘upwind’ finite elements. International Journal for Numerical Methods in Engineering, 12, 1359–1365.CrossRefzbMATHGoogle Scholar
  58. 133.
    Hughes, T. J. R. (1987). The finite element method: Linear static and dynamic finite element analysis. Englewood Cliffs, NJ: Prentice Hall, Inc. With the collaboration of Robert M. Ferencz and Arthur M. Raefsky. MR1008473 (90i:65001).Google Scholar
  59. 134.
    Hughes, T. J. R. (1995). Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Computational Methods in Applied Mechanical Engineering, 127(1–4), 387–401. doi:10.1016/00457825(95)00844-9. MR1365381 (96h:65135).Google Scholar
  60. 135.
    Hughes, T. J. R., Feijóo, G. R., Mazzei, L., & Quincy, J.-B. (1998). The variational multiscale method—a paradigm for computational mechanics. Computational Methods in Applied Mechanical Engineering, 166(1–2), 3–24. doi:10.1016/S0045-7825(98)00079-6. MR1660141 (99m:65239).Google Scholar
  61. 136.
    Hughes, T. J. R., Franca, L. P., & Hulbert, G. M. (1989). A new finite element formulation for computational fluid dynamics, VIII: The Galerkin/least-squares method for advective-diffusive equations. Computational Methods in Applied Mechanical Engineering 73(2), 173–189. doi:10.1016/0045-7825(89)90111-4. MR1002621 (90h:76007).Google Scholar
  62. 137.
    Hughes, T. J. R., & Sangalli, G. (2007). Variational multiscale analysis: The fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM Journal of Numerical Analysis, 45(2), 539–557. doi:10.1137/050645646. MR2300286 (2008c:65332).Google Scholar
  63. 139.
    Johnson, C. (1987). Numerical solution of partial differential equations by the finite element method. Cambridge: Cambridge University Press. MR925005 (89b:65003a).Google Scholar
  64. 141.
    Knabner, P., & Angermann, L. (2003). Numerical methods for elliptic and parabolic partial differential equations. Texts in Applied Mathematics (Vol. 44). New York: Springer. MR1988268 (2004j:65002).Google Scholar
  65. 144.
    Kozubek, T., Vondrák, V., Menšık, M., Horák, D., Dostál, Z., Hapla, V., et al. (2013). Total FETI domain decomposition method and its massively parallel implementation. Advances in Engineering Software, 60–61, 14–22. doi:10.1016/j.advengsoft.2013.04.001.CrossRefGoogle Scholar
  66. 145.
    Kreyszig, E. (1989). Introductory functional analysis with applications. Wiley Classics Library. New York: Wiley. MR992618.Google Scholar
  67. 147.
    Lax, P. D. (2002). Functional analysis. Pure and Applied Mathematics (New York). New York: Wiley Interscience. MR1892228.Google Scholar
  68. 148.
    Lions, J.-L., & Magenes, E. (1972). Non-homogeneous boundary value problems and applications (Vol. I). New York: Springer. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR0350177 (50#2670).Google Scholar
  69. 149.
    Madureira, A. L. (2009). A multiscale finite element method for partial differential equations posed in domains with rough boundaries. Mathematics of Computation, 78(265), 25–34. MR2448695.Google Scholar
  70. 150.
    Madureira, A. L. (2015). Abstract multiscale-hybrid-mixed methods. Calcolo, 52(4), 543–557. MR3421669.Google Scholar
  71. 154.
    Målqvist, A. (2011). Multiscale methods for elliptic problems. Multiscale Modelling and Simulation, 9(3), 1064–1086. doi:10.1137/090775592. MR2831590 (2012j:65419).Google Scholar
  72. 155.
    Målqvist, A., & Peterseim, D. (2014). Localization of elliptic multiscale problems. Mathematics of Computation, 83(290), 2583–2603. doi:10.1090/S0025-5718-2014-02868-8. MR3246801.MathSciNetCrossRefzbMATHGoogle Scholar
  73. 157.
    McLean, W. (2000). Strongly elliptic systems and boundary integral equations. Cambridge: Cambridge University Press. MR1742312.Google Scholar
  74. 161.
    Ming, P., & Yue, X. (2006). Numerical methods for multiscale elliptic problems. Journal of Computational Physics, 214(1), 421–445. MR2208685 (2006j:65359).Google Scholar
  75. 163.
    Nečas, J. (1967). Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague (French). MR0227584 (37 #3168).Google Scholar
  76. 169.
    Pian, T., & Tong, P. (1969). Basis of finite element methods for solid continua. International Journal for Numerical Methods in Engineering, 1, 3–28.CrossRefzbMATHGoogle Scholar
  77. 171.
    Quarteroni, A., & Valli, A. (1994). Numerical approximation of partial differential equations. Springer Series in Computational Mathematics (Vol. 23). Berlin: Springer. MR1299729 (95i:65005).Google Scholar
  78. 173.
    Raviart, P.-A., & Thomas, J. M. (1977). Primal hybrid finite element methods for 2nd order elliptic equations. Mathematics of Computation, 31(138), 391–413. MR0431752 (55 #4747).Google Scholar
  79. 181.
    Sangalli, G. (2003). Capturing small scales in elliptic problems using a residual-free bubbles finite element method. Multiscale Modelling and Simulation, 1(3), 485–503 (electronic). MR2030161 (2004m:65202).Google Scholar
  80. 185.
    Toselli, A., & Widlund, O. (2005). Domain decomposition methods—algorithms and theory. Springer Series in Computational Mathematics (Vol. 34). Berlin: Springer. MR2104179 (2005g:65006).Google Scholar
  81. 192.
    Wang, W., Guzmán, J., & Shu, C.-W. (2011). The multiscale discontinuous Galerkin method for solving a class of second order elliptic problems with rough coefficients. International Journal of Numerical Analysis and Modeling, 8(1), 28–47. MR2740478 (2012a:65346).Google Scholar
  82. 193.
    Weinan E, & Engquist, B. (2003). Multiscale modeling and computation. Notices of the American Mathematical Society, 50(9), 1062–1070. MR2002752 (2004m:65163).Google Scholar
  83. 199.
    Weinan E, Ming, P., & Zhang, P. (2005). Analysis of the heterogeneous multiscale method for elliptic homogenization problems. Journal of the American Mathematical Society, 18(1), 121–156 (electronic). MR2114818 (2005k:65246).Google Scholar

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© The Author(s) 2017

Authors and Affiliations

  • Alexandre L. Madureira
    • 1
    • 2
  1. 1.Laboratório Nacional de Computação Científica-LNCCPetrópolisBrazil
  2. 2.Fundação Getúlio Vargas-FGVRio de JaneiroBrazil

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