Syllabus

Preliminaries¶

• Recap on Elements of Functional Analysis: Introduction to Lebesgue Spaces, Sobolev Spaces, and key inequalities. Applied examples demonstrating their relevance in solving real-world problems.

• Recap on Elements of Numerical Analysis: Basics of interpolation, quadrature, tensorization, and their importance in FEM. Introduction to meshes and linear algebra essentials for FEM.

Weak Formulations and Well-Posedness¶

• Continuous Properties: Weak Formulation of Model Problems, Lax–Milgram Lemma and Banach–Nečas–Babuška (BNB) Theorem.

• Galerkin Approximation: Understanding discrete well-posedness, Cèa’s Lemma, and basic error estimates.

Introduction to Finite Elements¶

• Local Finite Elements: Detailed study of finite elements as triples and quadruples, shape-regularity for affine meshes, and practical workshops using FEM software.

• Finite Element Interpolation: Deny-Lion Lemma, Bramble–Hilbert Lemmas and interpolation errors, including software implementation sessions.

Inverse and Functional Inequalities¶

• Local and Global Inequalities: inverse inequalities in cells and on faces, understanding broken spaces, jumps, and functional inequalities in meshes.

A priori Error Analysis¶

• Model Problem and Error Analysis: Elliptic problems with Dirichlet/Robin/Neumann boundary conditions, elliptic regularity, and error analysis in $H^1$ and $L^2$.

Stabilization Mechanisms¶

• Diffusion-Transport-Reaction Equations: well-posedness, streamline diffusion, GLS, and SUPG, illustrating their necessity and effectiveness.

A-posteriori Error Analysis and Adaptivity¶

• Error Estimation and Adaptivity: Implementing adaptive mesh refinement strategies based on a-posteriori error estimates in programming assignments.

Variational Crimes¶

• Error Analysis for Variational Crimes: Analysis of numerical integration effects and Strang’s Lemmas

Discontinuous Galerkin¶

• Nitsche’s Method and SIP: discontinuous Galerkin methods, stability, well-posedness, and error analysis.

Abstract Saddle Point Problems¶

• Saddle Point Problems and Mixed Finite Element Approximation: Babuška–Brezzi Theorem, mixed formulations, and mixed finite element approximations.

Darcy and Stokes¶

• Application Modules: Specialized modules on Darcy and Stokes equations, focusing on mixed formulations, $H(div)$ finite elements, and stable pairs for Stokes equations, culminating in a collaborative project.

Projects and Seminars¶

• Students presentations: Software projects (possibly in groups), seminars, and presentations from students.

Course Features¶

• Flipped Classrooms and Laboratories: For practical sessions on Python and C++ implementations.

• Peer Review Sessions: Encourages collaborative learning and exposure to diverse problem-solving techniques.

• Update and Feedback Mechanisms: Incorporates the latest research developments and adjusts content based on student feedback.

Assessment¶

• Projects: Real-world problem solving using FEM.

• Practical Implementations: Software-based assignments.

• Final Exam: Covers theoretical and practical aspects of the course. It includes an oral examination (mandatory), possibly a software project, or a seminar on a topic to be discussed with the teacher.

Learning Outcomes¶

Upon completion, students will have an advanced understanding of FEM for solving PDEs, capable of applying this knowledge to complex problems across various domains, and proficient in using FEM software and programming languages for numerical analysis.