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Advanced PDE modeling : analysis & FE method

Presentation

Boundary value problems : weak solution vs classical solution. Sobolev spaces, Stampacchia theorem, Lax-Milgram. Green’s formulae. Boundary conditions.

Energy minimization (symmetric case).

Finite Element method: discretisation, approximation, implementation.

Error analysis (a-priori).

Assessment method of schemes and computational codes.

Error analysis (a-posteriori), mesh refinement.

Advection-diffusion equation (SUPG stabilisation).

Unsteady models: semi-discretisation.

Stokes system: discrète inf-sup condition, FE schemes.

Elasticity system: introduction, schemes.

Practical : assembly algorithm ; scheme assessments (accuracy) ; mesh refinement. Python-Fenics.

Objectives

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

 

-          How to write the weak (variational) form of the classical PDE models (with the corresponding energy minimization, symmetric case).

-          Write and implement a finite element scheme

-          Define basic a-posteriori estimators

-          Use a finite element library

Needed prerequisite

Fundamentals of PDE models, math. analysis,

Basic numerical methods

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...