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An Introduction to the

Finite Element Method (FEM)

for Differential Equations in 1D

Mohammad Asadzadeh

June 24, 2015

Contents1 Introduction1

1.1 Ordinary differential equations (ODE) . . . . . . . . . . . . . 1

1.2 Partial differential equations (PDE) . . . . . . . . . . . . . . . 2

1.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Polynomial Approximation in 1d 9

2.1 Overture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Basis function in nonuniform partition . . . . . . . . . 14

2.2 Variational formulation for (IVP) . . . . . . . . . . . . . . . . 17

2.3 Galerkin finite element method for (2.1.1) . . . . . . . . . . . 19

2.4 A Galerkin method for (BVP) . . . . . . . . . . . . . . . . . . 21

2.4.1 The nonuniform version . . . . . . . . . . . . . . . . . 26

2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Interpolation, Numerical Integration in 1d 31

3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Lagrange interpolation . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Numerical integration, Quadrature rules . . . . . . . . . . . . 41

3.3.1 Composite rules for uniform partitions . . . . . . . . . 44

3.3.2 Gauss quadrature rule . . . . . . . . . . . . . . . . . . 48

4 Two-point boundary value problems 53

4.1 A Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 The finite element method (FEM) . . . . . . . . . . . . . . . . 58

4.3 Error estimates in the energy norm . . . . . . . . . . . . . . . 59

4.4 FEM for convection-diffusion-absorption BVPs . . . . . . . . 65

4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

iii ivCONTENTS

5 Scalar Initial Value Problems 81

5.1 Solution formula and stability . . . . . . . . . . . . . . . . . . 82

5.2 Finite difference methods . . . . . . . . . . . . . . . . . . . . . 83

5.3 Galerkin finite element methods for IVP . . . . . . . . . . . . 86

5.3.1 The continuous Galerkin method . . . . . . . . . . . . 87

5.3.2 The discontinuous Galerkin method . . . . . . . . . . . 90

5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Initial Boundary Value Problems in 1d 95

6.1 Heat equation in 1d . . . . . . . . . . . . . . . . . . . . . . . . 95

6.1.1 Stability estimates . . . . . . . . . . . . . . . . . . . . 96

6.1.2 FEM for the heat equation . . . . . . . . . . . . . . . . 100

6.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 The wave equation in 1d . . . . . . . . . . . . . . . . . . . . . 106

6.2.1 Wave equation as a system of PDEs . . . . . . . . . . 107

6.2.2 The finite element discretization procedure . . . . . . 108

6.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A Answers to Exercises115

B Algorithms and MATLAB Codes 121

Table of Symbols and Indices135

CONTENTSv

Preface and acknowledgments.This text is an elementary approach to finite element method used in numerical solution of differential equations in one space dimension. The purpose is to introduce students topiecewise poly- nomial approximation of solutions using a minimum amount oftheory. The presented material in this note should be accessible to students with knowl- edge of calculus of single- and several-variables and linear algebra. The theory is combined with approximation techniques that are easily implemented by

Matlab codes presented at the end.

During several years, many colleagues have been involved inthe design, presentation and correction of these notes. I wish to thank Niklas Eriksson and Bengt Svensson who have read the entire material and mademany valu- able suggestions. Niklas has contributed to a better presentation of the text as well as to simplifications and corrections of many key estimates that has substantially improved the quality of this lecture notes. Bengt has made all xfigfigures. The final version is further polished by John Bondestam Malm- berg and Tobias Geb¨ack who, in particular, have many usefulinput in the

Matlab codes.

viCONTENTS void Chapter 1IntroductionIn this lecture notes we present an introduction to approximate solutions for differential equations. A differential equation is a relation between a function and its derivatives. In case the derivatives that appear in adifferential equa- tion are only with respect to one variable, the differential equation is called ordinary. Otherwise it is called a partial differential equation. For example, du dt-u(t) = 0,(1.0.1) is an ordinary differetial equation, whereas ∂u ∂t-∂2u∂x2= 0,(1.0.2) is a partial differential (PDE) equation. In (1.0.2) ∂u ∂t,∂2u∂x2denote the partial derivatives. Heretdenotes the time variable andxis the space variable. We shall only study one space dimentional equations that are either stationary (time-independent) or time dependent. Our focus will be on the following equations:

1.1 Ordinary differential equations (ODE)

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