An Introduction to the Finite Element Method for Differential Equations. Mohammad Asadzadeh
(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: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.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).
Example 26.1
As a simple example of an ODE, we mention the population dynamics model
(1.1.1)
If
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 equationsandare 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 areTo 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
(1.2.1)
(1.2.2)
(1.2.3)
where we have the space variable
Often, the dimension is obvious from the context and therefore, usually, the subindex
