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Advanced Finite Element Analysis

Course Type: 
PhD Course
Academic Year: 
March - May
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 programA priori estimates (4h)

  • Lax Milgram Lemma
  • Cea’s Lemma
  • Bramble Hilbert Lemma
  • Inverse estimates
  • Trace estimates

Stabilization mechanisms (2h)

  • Diffusion-Transport-Reaction equations
  • Strongly consistent stabilizations (Galerkin Least Square (GLS), and Streamline Upwind Petrov Galerkin (SUPG) methods)

Mesh adaptivity and a posteriori estimates (2h)

  • a posteriori error estimates
  • solve, estimate, mark, refine

Non-conforming finite element methods (4h)

  • Strangs I and II Lemmas
  • Symmetric Interior Penalty method
  • Analysis of SIP DGFEM

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)

Non matching discretisation techniques (2h)

  • Immersed Boundary Method 
  • Dirac Delta Approximation
  • Immersed Finite Element Method

 Short seminars from students, valid as exam (2h)

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