## 6 Numerical Methods and Algorithms
### Models at the Macro-Scale
#### 6.1.1 Discretization in the Space-Time Domain  
Numerical techniques called the finite-element, finite-difference, and finite-volume methods (FEM, FDM, and FVM, respectively) are used to discretize Partial Differential Equations (PDEs) into the space domain. Nevertheless, they differ from one another in different ways, and each of them has both advantages and disadvantages. Understanding these distinctions could help to select the method most appropriate for the purposes. In the following, a description of the ideas on the basis of each of those methods is given. See [127; 128; 129] for other discussions about finite elements versus finite volumes versus finite differences. Finite ElementsA strong computational method for solving differential and integral equations that come up in many applied scientific and engineering domains is the Finite Element Method (FEM) [125]. The fundamental idea behind the FEM is to consider a given domain as a collection of basic geometric shapes called finite elements, for which the approximation functionsrequired to solve a differential equation can be generated systematically. Indeed, the solution \(u\) of a differential equation can be approximated, on each element \(e\), by a linear combination of unknown parameters \(c_{j}\) and appropriately selected functions \(\phi_{j}\):  
\[u^{h}(x)=\sum_{i=1}^{l}c_{i}\phi_{i}(x),\text{ on }\Omega^{e}. \tag{85}\]  
For a given differential equation, it is possible to develop different finite element models, depending on the choice of a particular type of approximation method (e.g., Galerkin, weak-form Galerkin, least-squares, subdomain, collocation, and so on). The finite element model is a set of algebraic relations among the unknown parameters \(c_{j}\) of the approximation: so, solving the problem to find an approximation \(u^{h}\) of \(u\) requires the solution of a system of algebraic equations with the unknown parameters \(c_{j}\). The major steps in the finite element formulation and analysis of a typical problem are as follows. 1. Discretization of the domain into a set of selected finite elements. This is accomplished by subdividing the given domain \(\tilde{\Omega}=\Omega\cup\Gamma\) into a set of subdomains \(\tilde{\Omega}^{e}=\Omega^{e}\cup\Gamma^{e}\) where \(\Gamma=\partial\Omega\) and \(\Gamma^{e}=\partial\Omega^{e}\), called finite elements (see Figure 8a). The phrase \"finite element\" often refers to both the geometry of the element and degree (or order) of approximation: The form of the element \(\Omega^{e}\) can be either triangle or quadrilateral, and the degree of interpolation over it can take on various forms, including linear and quadratic. The finite element mesh of the domain \(\Omega\) is the non-overlapping total (or assembly) of all elements used to represent the actual domain, and it is represented by \(\Omega_{h}\). In general, \(\Omega_{h}\) may not equal the actual domain \(\Omega\) because of its potentially complex geometry. Nodal points, also known as nodes, are typically taken at appropriate positions in the structure, usually to simplify the element. They are used to characterize the individual elements and subsequently the complete mesh structure (see Figure 8b). 2. Construction of a statement, often a weighted-integral in a weak-form statement according to Weighted Residual Methods (WRMs) [130], which is equivalent (in some sense) to the differential equation to be analyzed over a typical element. A general class of techniques, called the Weighted Residual Method, was created to obtain an approximate solution to the problem of the form \[L(u(x))=f(x),\text{ on }\Omega\] (86) where \(L(u)\) is a general linear differential operator.