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An Introduction to Recent Developments in Numerical Methods for Partial Differential Equations

  • Daniele Antonio Di Pietro
  • Alexandre Ern
  • Luca Formaggia
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 15)

Abstract

Numerical Analysis applied to the approximate resolution of Partial Differential Equations (PDEs) has become a key discipline in Applied Mathematics. One of the reasons for this success is that the wide availability of high-performance computational resources and the increase in the predictive capabilities of the models have significantly expanded the range of possibilities offered by numerical modeling.

Novel discretization methods, the solution of ill-posed and nonlinear problems, model reduction and adaptivity are main topics covered by the contributions of this volume. This introductory chapter provides a brief overview of the book and some related references.

References

  1. 1.
    Ainsworth, M., Wajid, H.A.: Optimally blended spectral-finite element scheme for wave propagation and nonstandard reduced integration. SIAM J. Numer. Anal. 48(1), 346–371 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. (N.S.) 47(2), 281–354 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bochev, P., Hyman, J.M.: Principles of mimetic discretizations of differential operators. In: Arnold, D., Bochev, P., Lehoucq, R., Nicolaides, R.A., Shashkov, M. (eds.) Compatible Spatial Discretization. The IMA Volumes in Mathematics and Its Applications, vol. 142, pp. 89–120. Springer, New York (2005)CrossRefGoogle Scholar
  4. 4.
    Boffi, D., Cavallini, N., Gastaldi, L.: The finite element immersed boundary method with distributed Lagrange multiplier. SIAM J. Numer. Anal. 53(6), 2584–2604 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burman, E.: Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: elliptic equations. SIAM J. Sci. Comput. 35(6), A2752–A2780 (2013)zbMATHGoogle Scholar
  6. 6.
    Burman, E.: Stabilised finite element methods for ill-posed problems with conditional stability. In: Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 114, pp. 93–127. Springer, Cham (2016)Google Scholar
  7. 7.
    Carstensen, C., Feischl, M., Page, M., Praetorius, D.: Axioms of adaptivity. Comput. Math. Appl. 67(6), 1195–1253 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Christiansen, S.H., Munthe-Kaas, H.Z., Owren, B.: Topics in structure-preserving discretization. Acta Numer. 20, 1–119 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cockburn, B., Di Pietro, D.A., Ern, A.: Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: Math. Model Numer. Anal. 50(3), 635–650 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cockburn, B., Fu, G., Sayas, F.J.: Superconvergence by M-decompositions. Part I: general theory for HDG methods for diffusion. Math. Comput. 86(306), 1609–1641 (2017)zbMATHGoogle Scholar
  12. 12.
    Di Pietro, D.A., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Di Pietro, D.A., Ern, A., Lemaire, S.: An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Mech. Eng. 14(4), 461–472 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Girault, V., Glowinski, R.: Error analysis of a fictitious domain method applied to a Dirichlet problem. Jpn. J. Ind. Appl. Math. 12(3), 487–514 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hughes, T.J.R., Evans, J.A., Reali, A.: Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput. Methods Appl. Mech. Eng. 272, 290–320 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)CrossRefGoogle Scholar
  17. 17.
    Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Prud’homme, C., Rovas, D.V., Veroy, K., Patera, A.T.: A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. M2AN Math. Model. Numer. Anal. 36(5), 747–771 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Arch. Comput. Meth. Eng. 15(3), 229–275 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Verfürth, R.: A posteriori error estimation techniques for finite element methods. In: Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Daniele Antonio Di Pietro
    • 1
  • Alexandre Ern
    • 2
    • 3
  • Luca Formaggia
    • 4
  1. 1.Institut Montpelliérain Alexander Grothendieck, CNRSUniversité MontpellierMontpellierFrance
  2. 2.Université Paris EstCERMICS (ENPC)Marne la ValléeFrance
  3. 3.INRIAParisFrance
  4. 4.MOXDipartimento di Matematica, Politecnico di MilanoMilanoItaly

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