Mathematics

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)

References

Articles

• 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.

Books

• 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