1
Introduction

There are two ways of spreading light:

to be the candle or the mirror that reflects it.

Edith Wharton

This book presents an introduction to the Galerkin finite element method (FEM) as a powerful and general tool for approximating solution of differential equations. Our objective is twofold.

1. i) To present the main ordinary and partial differential equations (ODEs and PDEs) modeling different phenomena in science and engineering and introduce mathematical tools and environments for their analytic and numerical studies.
2. ii) To construct some common FEMs for approximate solutions of differential equations and analyze their well-posedness (existence, uniqueness, and stability of such approximate solutions) as well as the accuracy of the approximation.

In its final step, a finite element procedure yields a linear system of equations (LSEs) where the unknowns are the approximate values of the solution at certain nodes. Then, an approximate solution is constructed by adapting piecewise polynomials of certain degree to these, approximate, nodal values.

The entries of the coefficient matrix and the right-hand side of FEM's final LSEs consist of integrals which, e.g. for complex geometries or less smooth, and/or more complex, data, are not always easily computable. Therefore, numerical integration and quadrature rules are introduced to approximate such integrals. Furthermore, iteration procedures are included in order to efficiently compute the numerical solutions of such obtained LSEs.

Interpolation techniques are presented for both accurate polynomial approximations and also to derive basic a priori and a posteriori error estimates necessary to determine qualitative properties of the approximate solutions. That is to show how the approximate solution, in some adequate measuring environment, e.g. a certain norm, approaches the exact solution as the number of nodes, hence, the number of unknowns increases. For convenience, the frequently used classical inequalities, such as the Cauchy-Schwarz' and Poincare, likewise the inverse and trace estimates, that are of vital importance in error analysis and stability estimates, are introduced. In the theoretical abstraction, we demonstrate the fundamental solution approach based on Green's functions and prove the Riesz (Lax-Milgram) theorem which is essential in proving the existence of a unique solution for a minimization problem that in turn is equivalent both to a variational formulation as well as a corresponding boundary value problem (BVP).

Galerkin's method for solving a general differential equation is based on seeking an approximate solution, which is

1. Easy to differentiate and integrate
2. Spanned by a set of "nearly orthogonal" base functions in a finite-dimensional vector space.
3. Satisfies Galerkin orthogonality relation.

   Roughly speaking, this means a closeness relation in the sense that:

   1. I). In a priori case, the difference between the exact and approximate solution is orthogonal to the finite dimensional vector space of the approximate solution.
   2. II). In a posteriori case, the residual of the approximate solution (=the difference between the left- and right-hand side of an expression obtained from the differential equation where exact solution is replaced by the approximate solution) is orthogonal to the finite dimensional vector space of the approximate solution.

1.1 Preliminaries

In this section, we give a brief introduction to some key concepts in differential equations. A standard classification and some general properties are presented in Trinities below.

• A differential equation is a relation between an unknown function and its derivatives.
• If the differentiation in the equation is with respect to only one variable, e.g. , (or in ), then the equation is called an ordinary differential equation (ODE).

• The order of a differential equation is the order of the highest derivative of the function that appears in the equation.
• If the function depends on more than one variable, and the differential equation possesses derivatives with respect to at least two variables, then the differential equation is called a partial differential equation (PDE), e.g. is a homogeneous PDE of the second order, whereas for , the equations and are nonhomogeneous PDEs of the second order.
• A solution to a differential equation is a function; (e.g. , , or above), which satisfies the corresponding differential equation.
• In general, the solution of a differential equation cannot be expressed in terms of elementary functions, and numerical methods are the only way to solve the differential equations through constructing approximate solutions. Then, the main questions are
  • To what extent does the approximate solution preserve the physical properties of the exact solution, or satisfies a corresponding, discrete, version of the differential equation (consistency)?
  • How sensitive is the solution to the change of the data (stability)?
  • How close is the approximate solution to the exact solution (convergence)?
  • Which are the adequate environments to measure this closeness?
• These are some of the questions that we want to deal within this text when approximating with the FEMs.
• A linear ODE of order has the general form: where denotes the derivative, with respect to , and , with (the -th order derivative). The corresponding linear differential operator is denoted by

1.2 Trinities for Second-Order PDEs

Problems modeled by PDEs of the second order can be classified using, the so-called, trinities. Below we introduce basic ingredients of this concept. For detailed study see, e.g. [68].

The usual three operators in PDEs of second order in .

where we have the space variable , the time variable , and denotes the second partial derivative with respect to . We also define a first-order operator, namely the gradient operator which is the vector valued operator

Often, the dimension is obvious from the context and therefore, usually, the subindex is suppressed and the operators and are simply denoted by (or by ) and , respectively.

Classifying general second-order PDEs in two dimensions.

1. I) The constant coefficients case

   A second-order PDE in two dimensions, with constant coefficients, can be written in its general form as

   We introduce the discriminant : a quantity that specifies the role of the coefficients of the second-order terms, in determining the equation type in the sense that the equation is

   Elliptic: if     Parabolic: if and Hyperbolic: if

2. II) The case of variable coefficients

   In the variable coefficients case, one can only have a local classification.

   Figure 1.1 Tricomi equation: an example of a variable coefficient classification.

The usual three types of problems in differential equations.

1. Initial value problems (IVPs)

   The simplest differential equation is for . But for any such , also for any constant . To determine a unique solution, a specification of the initial value is generally required. For example for , we have and the general solution is . With an initial value of , we get . Hence, the unique solution to this IVP is . Likewise, for a time-dependent differential equation of second order (two time derivatives), the initial values for , i.e. and , are generally required. For a PDE such as the heat equation, the initial value can be a function of the space variable.

2. Boundary value problems (BVP) )

   Consider the one-dimensional stationary heat equation:

   In order to determine a solution uniquely (see Remark 1.2), the differential equation is complemented by boundary conditions imposed at the boundary points and ; for example and , where and are given real numbers.

3. Eigenvalue problems (EVPs)

   Let be a given square, say matrix. The relation , is a linear equation system, where is an eigenvalue and is an eigenvector. In the Example 1.7 below, we introduce the corresponding terminology for differential...