 # 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.
• 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...