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

Non-conforming finite element methods (2h)

• 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)

Fluid structure interaction problems (2h)

• Problem definition
• Classical Finite Element Approximation: Arbitrary Lagrangian Eulerian Formulation

Immersed Boundary Method (2h)

• Definition
• Finite Difference Approximation
• Dirac Delta Approximation

Immersed Finite Element Method (4h)

• Definition
• Analysis of IFEM - 1D simple case
• Stability of IFEM (2D and 3D case)

Student projects (4h)

• Short seminars from students, valid as exam