# Foundations of the Finite Element Method SPRING 2020

#### Basic Information

• Instructor: Manil Suri, Math/Psych 419, x52311, suri@umbc.edu,
office hours: MW 3:30-4:30 or by appointment
• Lectures: MW 5:30-6:45, MP 105
• Reference Tex (primary)t: Introduction to Automated Modeling with FEniCS by Ridgway Scott
• Reference Text: Finite Elements by Dietrich Braess (3rd edition, previous eds OK)
• Reference Text: Finite Element Analysis by Barna Szabo and Ivo Babuska (We will refer to this for ‘p’ and ‘hp’ methods)
• Reference Text: Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson (Dover Edition) (We will refer to Chapters 8 and 9 for parabolic and hyperbolic problems)
• Prerequisites: A good background in mathematical and numerical analysis will be advantageous. Programming will be done in Matlab (or language of your choice).

#### Overview

Finite element methods are used to approximate the solutions of partial differential equations which arise in various engineering and other applications. The emphasis of this course will be on the theoretical foundations of the method.

We will start with the 1-d case, and go through it in detail (culminating in you writing a 1-d program). Then we will proceed to the 2-d case, using Chapters 1-7 of Scott, with additional material from book by Braess. This will include the description, error analysis and some illustrative examples with traditional ‘h’ type methods (we will review Chapter 30 by Scott, on Sobolev Spaces, as needed).

Following this will be a brief treatment of ‘p’ and ‘hp’ type methods (reference is book by Szabo and Babuska). Special attention will be paid to the approximation of singularities (such as those that develop at cracks). We will then cover a selection of topics that may include nonlinear problems, Stokes problem, a posteriori error estimators, mixed methods (much of this will be taken from Scott, but some will be from Braess). In particular, we will cover some parabolic and hyperbolic problems (for which additional material will be from Chapters 8 and 9 of the book by Johnson).

There will be one project, which will be to write a finite element code in one dimension, and use it to investigate some of the theory developed. Note that although Scott’s book is based on the finite element code FEniCS, we will generally use Matlab’s PDE Modeler for illustrative computations.

#### Learning Goals

This course has the following learning goals.

• Knowledge of the mathematical theory of variational procedures and the finite element method. The idea is to have an understanding behind the “black box” of FEM codes. (Text readings and homework problems will assist in realizing this goal.)
• Theoretical and practical understanding of convergence rates. (Text, homework and project work will assist in realizing this goal.)
• Learning the architecture of FEM programs by writing a one-dimensional code. (The project will realize this goal.)
• Familiarity with some of the applications of finite elements. (These topics will be chosen according to the interests of the class.)