Lecture Notes: The Galerkin Method

TL;DR

This paper introduces the Galerkin method for approximating solutions to PDEs and integral equations, covering theoretical foundations, computational examples, and an application to nonlinear boundary value problems.

Contribution

It provides a comprehensive introduction to the Galerkin method, including analysis background, computational implementation, and a proof of existence for nonlinear problems.

Findings

Demonstrates the Galerkin method's effectiveness through computational examples

Provides MATLAB code for practical implementation

Proves existence of solutions for a nonlinear boundary value problem

Abstract

These lecture notes introduce the Galerkin method to approximate solutions to partial differential and integral equations. We begin with some analysis background to introduce this method in a Hilbert Space setting, and subsequently illustrate some computational examples with the help of a sample matlab code. Finally, we use the Galerkin method to prove the existence of solutions of a nonlinear boundary value problem.