MAGIC098: Adaptive Finite Element Methods

Adaptive Finite Element Methods is an advanced course in numerical analysis. 
The first aim is to provide an introduction to the finite element method (FEM) itself, which is one of the most powerful methods for approximating solutions to partial differential equations (PDEs). 
The second aim is to provide an introduction to adaptive finite element methods, which are techniques that generate optimal meshes, and thereby provide the most efficient methodology to approximate solutions to PDEs.
Learning objectives:
* Discretise PDEs with FEM
* Analyse the stability and convergence of FEM
* Estimate errors of approximations to PDEs
* Approximate PDEs using adaptive mesh refinement
* Analyse convergence of adaptive methods
* Introduce Neural Network (deep learning) approximations of PDEs
Within this course, there is a mix of methods, theory and algorithms. 

Students are expected to have had an introductory course in numerical analysis. They should also have basic knowledge of partial differential equations (PDEs), and some programming experience.
(It is not assumed students are familiar with the finite element method itself.)

• Intro FEM I: Weak formulations of PDEs in 1-D
• Intro FEM II: FEM discretization in 1-D
• Intro FEM III: Well-posedness of weak formulations of PDEs
• Intro FEM IV: FEM discretization in 2-D
• Intro FEM V: Convergence of FEM for PDEs
• Adaptive FEM I: Computable error estimates for FEM
• Adaptive FEM II: Adaptive FEM for PDEs and its convergence
• Adaptive FEM III: Optimality of adaptive FEM for PDEs
• Adaptive FEM IV & V: Neural networks and deep learning for PDEs

Assessment:

• Set of problems similar to examples in Lectures.

Recommended readings supporting lectures:

• [Larson, Bengzon, 2013] The Finite Element Method: Theory, Implementation, and Applications, Springer Book, Link to preprint 
• [Johnson, 1987] Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press Book, Link
• [Ern, Guermond, 2021 b] Finite Elements II: Galerkin Approximation Elliptic and Mixed PDEs, Springer Book, Link to preprint
• [Ainsworth, Oden, 2000] A Posterori Error Estimation in Finite Element Analysis, Wiley Interscience Book, Link
• [van der Zee, 2004] Goal-Adaptive Discretization of Fluid–Structure Interaction, PhD thesis, Link
• [Nochetto, 2010] Why Adaptive Finite Element Methods Outperform Classical Ones, Proceedings of the International Congress of Mathematicians, Link
• [E, Yu, 2018] The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems, Link to preprint
• [Higham, Higham 2019] Deep Learning: An Introduction for Applied Mathematicians, Link to preprint

Other references used in lectures:

• [Gander, Wanner, 2012] From Euler, Ritz, and Galerkin to Modern Computing, Link 
• [Brooks, Hughes, 1982] Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Link
• [Courant, 1943] Variational methods for the solution of problems of equilibrium and vibrations, Link
• [Cea, 1964] Approximation variationnelle des problemes aux limites, Link 
• [Ciarlet, 1978] The Finite Element Method for Elliptic Problems, Link
• [Babuska, Rheinboldt, 1978] Error Estimates for Adaptive Finite Element Computations, Link 
• [Reichenbach, 1951] The Rise of Scientific Philosophy, Link
• [Oden, Prudhomme, 2000] Goal-oriented error estimation and adaptivity for the finite element method, Link
• [Babuska, Vogelius, 1984] Feedback and adaptive finite element solution of one-dimensional boundary value problems, Link
• [Dorfler, 1996] A Convergent Adaptive Algorithm for Poisson’s Equation, Link
• [Binev, Dahmen, DeVore, 2004] Adaptive Finite Element Methods with convergence rates, Link
• [Strang, 2018] The Functions of Deep Learning, Link