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Advanced Finite Element Analysis

Course Type: 
PhD Course
Academic Year: 
May - June
20 h

An advanced course dedicate to the analysis of finite element methods, as found in modern numerical analysis literature. A basic knowledge of Sobolev spaces is expected.

Detailed program

  • Review of basic results for Finite Element Analysis (Galerkin method) (2h)
    • Lax Milgram Lemma, Cea’s Lemma, Bramble Hilbert Lemma
    • Inverse estimates, trace estimates
  • Petrov-Galerkin finite element methods (2h)
    • Ladyzhhenskaya, Brezzi, Babuska VS Lax Milgram
  • Mixed and hybrid finite element methods (4h)
    • Mixed Laplace Problem
    • Stokes Problem
    • A priori error estimates (exploiting Strang Lemmas)
    • Proving the inf-sup (Fortin’s trick, macroelement technique)
  • Stabilization mechanisms for Finite Element Methods (2h)
    • Diffusion-Transport-Reaction equations
    • Strongly consistent stabilizations (Galerkin Least Square (GLS), and Streamline Upwind Petrov Galerkin (SUPG) methods)
  • A posteriori error estimates (2h)
    • Residual based estimates
    • L2 a posteriori estimates
  • Variational Crimes, or Discontinuous Galerkin Methods
    • Analysis of Discontinuous Galerkin Methods
    • Stabilization of DG Methods
  • Analysis of Krylov Supspace Methods (Giuseppe Pitton)
  • Boundary Element Mehtods (Nicola Giuliani)



  • Babuška I. Error-bounds for finite element method. Numer Math
  • Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. … Numer Anal Mathématique …. 1974:129–151.
  • Stenberg R. A technique for analysing finite element methods for viscous incompressible flow. Int J Numer Methods Fluids. 1990;11(6):935–948. doi:10.1002/fld.1650110615
  • Brezzi F, Cockburn B, Marini LD, Süli E. Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput Methods Appl Mech Engrg. 2006;195(25-28):3293–3310.


  • Quarteroni A. Numerical Models for Differential Problems.; 2009.
  • Boffi D, Brezzi F, Fortin M. Mixed Finite Element Methods and Applications.; 2013. doi:10.1007/978-3-642-36519-5
  • Liesen J, Strakos Z. Krylov Subspace Methods: Principles and Analysis. 1st ed. Oxford University Press; 2012.
  • Sauter SA, Schwab C. Boundary element methods. New World Publ. 1992
A - 134 on May 24, 30, 31
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