Fundamental Concepts of Finite Element Method (FEM)
In this chapter the basic principles of FEM are systematically developed and reviewed for prototypical advection-dispersion equations (ADE’s). The different spatial and temporal discretization techniques are addressed. The important approximate solutions
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Fundamental Concepts of Finite Element Method (FEM)
8.1 Introduction In the previous Chaps. 3–5 the governing continuum balance equations in form of partial differential equations (PDE’s) have been derived for a wide range of flow, mass and heat transport processes in porous and fractured media. Their solution under given IC’s and BC’s, such as described in Chap. 6, requires appropriate and efficient mathematical methods, which can be firstly grouped into analytical and numerical methods. There is a family of powerful analytical methods (e.g., Fourier and Laplace transformation, complex variable techniques, Green’s functions, perturbation methods, power series), which are capable of solving a certain number of problems in an exact way. However, exact analytical solutions are often only attainable for elementary linear (or quasi-linear) problems on simple (regular) geometries. Very few analytical solutions exist for nonlinear problems with regions of regular geometry, however, these are usually approximate solutions in terms of an infinite series or some transcendental functions that can be evaluated only approximately. If exact analytical solutions are available on idealized problems they are often advantageous in comparison to numerical results for purposes of verification and estimation of errors arising in the alternative numerical methods. Problems involving irregular geometry, materials with variation in properties, nonlinear relationships and/or complex BC’s are intractable by analytical methods and numerical methods must be used in general. They allow the solution for a broad range of problems. The key feature of any numerical method is in the approximate solution of the basic PDE’s via spatial and temporal discretizations, in which the solution variables, which are basically continuous functions of space and time, are obtained by discrete values, defined at specific points in space and time (Fig. 8.1). In doing this approximation, the governing PDE’s are replaced by a number (often, a very large number) of linear (or linearized) algebraic equations, which can be easily solved via computers. As a consequence of the numerical approximation, errors are naturally inherent in the solution and the big challenge of numerical methods is to minimize these numerical errors and find best accurate, convergent and stable H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5 8, © Springer-Verlag Berlin Heidelberg 2014
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8 Fundamental Concepts of Finite Element Method (FEM) active cell
inactive cell
finite element
single well
domain and boundary
finite difference approach (structured grid)
finite element approach (unstructured mesh)
Fig. 8.1 Example of 2D domain discretized by finite differences and finite elements
solutions by using efficient, general and robust strategies of approximation. It is important to ensure that the approximation satisfies certain important properties of the exact solution, e.g., conservativity, boundedness and consistency (see Sect. 1.2.2 for further discussion). We can clas
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