Finite Element Methods & Model Reductions

Presentation

• Boundary value problems : weak solution vs classical solution; Lax-Milgram theory.
• Analysis of Laplace equation (with mixed b.c.); energy minimization (symmetric case); transmission conditions.
• FE Modeling practical with FreeFem++ (classical models in engineering).
• The Finite Element method: basic principles, implementation, (a-priori) error estimations. Convergence curves, assessment of computational codes.
• Advection-diffusion equation (SUPG stabilisation).
• Unsteady models: semi-discretisation. (Scalar) non linear problems and linearization.
• Error analysis (a-posteriori), mesh refinement.
• Constraints & Lagrangian multiplier. Stokes system, inf-sup condition, FE schemes. Elasticity system: introduction, schemes.
• FE Practical in Python: assembly algorithm, code assessments.
• Domain Decomposition Methods & preconditionners. Krylov subspace methods; common preconditonners; Schwarz methods. 
• DDM Practical in Freefem++.

Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

•  Develop weak (variational) forms of classical PDE models (with the corresponding energy minimization if symmetric pb).
• Detail FE schemes (Pk, Qk) and programming them.
• Develop Domain Decomposition algorithms.

Needed prerequisite

Fundamentals of PDE equations I4MMNP71

Numerical analysis I3MIMT11

Matrix computation I3MIMT31

Basics of functional analysis, differential calculus.

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...