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# 2.5 Finite Element Spaces and Meshes

The basic concepts of Galerkin's method, such as weak formulation of the problem and the transformation of this weak formulation from the functional space with infinite basis into the functional space with finite basis, as presented in Sections 2.3 and Sections 2.4, are presented here to introduce the finite element method.

The construction of the finite element space begins with subdividing the domain into a set of non-overlapping elements . The domain can now be approximated with a mesh domain, (2.21)

We denote as the set of all points of the mesh domain . Each point has an unique global index , where is the number of all points in the mesh. A point has several local indices.

The basis functions of the finite element space fulfill, (2.22)

Since they are defined on nodes of the mesh, we call them basis nodal functions. In the finite element praxis basis nodal functions are almost exclusively low order polynomials.

For further discussion it will be useful to have approximations of functions represented for each element with (2.23) represents the local index of the element vertices and is the number of element vertices.
We use linear basis functions and tethraedrons as elements. Therefore , and each of the functions , is approximated on the elements of the discretization using the linear basis functions , (2.24)

Now we can also define the operator for each element of the discretization, (2.25)

The inner product is calculated over the element . The local residuum vector is defined as,  (2.26)

Determining the operator requires the calculation of the basic nodal functions, (2.27)

The coefficients are functions of the nodal coordinates and is the volume of the element . Constructing of frequently demands calculations of the following integral, (2.28)

In this case (2.27) can be used, but instead it is more practical to project the discretization element (Figure 2.1) into normalized coordinate system element (Figure 2.2). Each point is a bijective projection of the corresponding point , (2.29)

The basis nodal functions on are , , and . (2.28) is now calculated as (2.30) is the Jacobian of the projection ( ), (2.31)      Next: 2.6 Newton Methods Up: 2. Finite Element Method Previous: 2.4 Time Dependent Problems

H. Ceric: Numerical Techniques in Modern TCAD