500679 - Finite Elements

courses

ID:

500679

Duration (hours):

76

CFU:

Located in:

PAVIA

Year:

2025

Date/time interval

Secondo Semestre (26/02/2026 - 12/06/2026)

Course Objectives

Being aware of the content and meaning of the basic theoretical results related to finite element methods. Having understood elementary concepts regarding the theoretical analysis of stability and error, the approximations by finite elements of problems in mixed variational formulation, the implementation aspects of the finite element method in MATLAB or Julia language. Knowing how to reproduce with awareness the main demonstrative phases of theory construction. Being able to frame and numerically solve some standard problems and types of elliptic differential equations.

Course Prerequisites

Fundamental notions of Analysis and Numerical Analysis

Teaching Methods

Lessons and computer lab practice

Assessment Methods

The exam consists of an oral test. This test aims to assess the level of understanding of the theoretical topics covered in class, the clarity of exposition, but also the ability to apply these notions in concrete situations. For this reason, the student will be required to have a substantial understanding of all the theory presented in class, as well as the implementation aspects, which can be assessed through questions on specific topics and through the proposal of problems related to the computer implementation of the finite element method. The questions will be articulated with varying difficulty to establish the depth of acquisition of these skills. The grading will be based on the overall breadth and depth of learning, as well as the clarity of exposition and the demonstrated problem-solving skills.

Texts

Teacher's notes.

A. Quarteroni, A. Valli: "Numerical Approximation of Partial Differential Equations", Springer-Verlag, 1994.

Daniele Boffi, Franco Brezzi, and Michel Fortin. Mixed finite element methods and applications. Berlin: Springer, 2013.

The aim of the course is to present the theoretical foundation of the finite element course, example of applications to the numerical solution of partial differential equations, and discuss its implementation. We will consider both diffusion (elliptic) problems and mixed problems, analyzing its stability, approximation properties. Then we will focus on mixed problems. In parallel, we will discuss and test its implementation in MATLAB language

Extended summary

Theory lessons will cover the following topics:
- fundamentals of Functional Analysis, with a particular emphasis on the W^{k,p} spaces and on primal variational formulations of elliptic problems
- approximation theory in Sobolev spaces: Deny-Lions Lemma and Brambe-Hilbert lemma
- Lagrange interpolation on n-simplices and corresponding interpolation error for Sobolev norms
-Galerkin method for elliptic problems and error estimates: Cea Lemma and duality techniques
- Finite Element Methods for elliptic problems, with particular emphasis to the bidimensional case
- mixed formulation of elliptic problems and its Galerkin discretization: existence, uniqueness, stability of the solution, and error analysis. Some example of Finite Elements for the diffusion problem in mixed form
- elasticity problem and its FEM discretization: the volumetric locking phenomenon and some possible cures

Computer Lab lessons will address the implementation of the finite element method, in MATLAB language. In particular:
- data structure and algorithm for the triangulation of a planar region
- interpolation and numerical integration of funtions on the triangulation
- local matrices and assembling
- Dirichlet and Neumann boundary condition
- finite element method for the Poisson problem in primal form with P1 elements
- implementation of the RT element
- finite element method for the Poisson problem in mixed form (Darcy problem)

REMARK: This is a tentative program. Significant changes might occur, also depending on the feedback provided by the Student during the lectures.