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2. Numerical Discretization Schemes

An important step in handling partial differential equations is to use and develop stable, consistent, and accurate algebraic replacements where most of the global/continuous information of the original problem and more importantly, the inherent structure, are retained. Several methods are currently in use, such as the finite volume (FV), finite element (FE), and finite difference (FD) methods, each with specific approaches to discretization. Topological equations have an intrinsically discrete nature, compared to the constitutive parts of the field equations, which are the central issues in the construction of effective discretization schemes and the only place where recourse to local representations is fully justified. Numerical discretization schemes can be briefly represented as a model reduction, e.g.:

$\displaystyle \mathcal{L}(u) = {\ensuremath{f}} \rightarrow \ensuremath{\mathbf{A}} \ensuremath{\mathbf{x}} = \ensuremath{\mathbf{b}}$ (2.1)

which transforms an infinite-dimensional operator equation into a finite-dimensional algebraic equation. Here it can already be seen that this is always accompanied by an inevitable loss of information due to the reduction of dimension. Briefly, the given discretization schemes address differently the task of replacing the partial differential equation system with algebraic ones. Therefore, generic discretization concepts, based on what has been called the reference discretization scheme [33,35], are introduced first. These concepts are then presented in the context of each of the other methods.




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R. Heinzl: Concepts for Scientific Computing