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 $\mathcal{V}_h$ begins with subdividing the domain $\Omega$ into a set $T_1,T_2,\dots,T_m$ of non-overlapping elements $(\forall i\neq j\Rightarrow T_i \cap T_j = \emptyset)$. The domain $\Omega$ can now be approximated with a mesh domain,

$$T_h(\Omega) = \underset{i=1,m}{\bigcup}\; T_i.$$ (2.21)

We denote as $P$ the set of all points of the mesh domain $T_h(\Omega)$. Each point $p_k\in P$ has an unique global index $k=1,\dots,N$, where $N$ is the number of all points in the mesh. A point has several local indices.

The basis functions $\varphi_i, i=1,\dots,N$ of the finite element space $\mathcal{V}_h$ fulfill,

$$\varphi_i(P_k)=\delta_{ik}, i,k=1,\dots,N.$$ (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 $\mathbf{c}$ represented for each element $T\in T_h(\Omega)$ with

$$\mathbf{c_e}=\Bigl(\sum_{i=1}^{N_p}\varphi_i c_{i1},\dots,\sum_{i=1}^{N_p}\varphi_i c_{iM}\Bigr)^T,$$ (2.23)

$i$ represents the local index of the element vertices and $N_p$ is the number of element vertices.
We use linear basis functions and tethraedrons $T\in T_h(\Omega)$ as elements. Therefore $N_p=4$, and each of the functions $c_i$, $i=1,\dots,M$ is approximated on the elements $T$ of the discretization $T_h(\Omega)$ using the linear basis functions $\varphi_j(\mathbf{x})$,

$$c_i(\mathbf{x})=\sum_{j=1}^4 c_{ij}\varphi_j(\mathbf{x}).$$ (2.24)

Now we can also define the operator $\ell^e : \Bbb{R}^{4\cdot M}\rightarrow \Bbb{R}^{4\cdot M}$ for each element of the discretization,

$$\ell_{ij}^e(c_{1,1},c_{1,2},c_{1,3},c_{1,4},\dots,c_{M,1},c_{M,2},c_{M,3},c_{M,4})= (\varphi_i,\mathcal{L}_j(\mathbf{c_e}))_T.$$ (2.25)

The inner product is calculated over the element $T$. The local residuum vector is defined as,

$$R=(R_{1,1},R_{1,2},R_{1,3},R_{1,4},\dots,R_{M,1},R_{M,2},R_{M,3},R_{M,4})^T,$$

$$R_{i,j}=\ell_{ij}^e(c_{1,1},c_{1,2},c_{1,3},c_{1,4},\dots,c_{M,1},c_{M,2},c_{M,3},c_{M,4})-(f_i,\varphi_j)_T.$$ (2.26)

Determining the operator $\ell^e$ requires the calculation of the basic nodal functions,

$$\varphi_k(\mathbf{x})=\varphi_k(x_1,x_2,x_3)=\frac{1}{3 V}(a_k x_1 + b_k x_2 + c_k x_3 + d_k).$$ (2.27)

The coefficients $a_k,b_k,c_k,d_k$ are functions of the nodal coordinates and $V$ is the volume of the element $T$ [11]. Constructing of $\ell^e$ frequently demands calculations of the following integral,

$$M_{pq}=\int_{T}\varphi_p(\mathbf{x}) \varphi_q(\mathbf{x}) d\Omega,\quad p,q=1,2,3,4.$$ (2.28)

In this case (2.27) can be used, but instead it is more practical to project the discretization element $T$ (Figure 2.1) into normalized coordinate system element $T_0$ (Figure 2.2). Each point $(x_1,x_2,x_3)\in T$ is a bijective projection of the corresponding point $(\xi,\eta,\zeta)\in T_0$,

$$\begin{split} x_1=& x_1^1 + (x_1^2-x_1^1)\xi + (x_1^3-x_1^1)... ..._3^1)\xi + (x_3^3-x_3^1)\eta + (x_3^4-x_3^1)\zeta. \end{split}$$ (2.29)

The basis nodal functions on $T_0$ are $\varphi^{0}_1(\xi,\eta,\zeta)=1-\xi-\eta-\zeta$, $\varphi^{0}_2(\xi,\eta,\zeta)=\xi$, $\varphi^{0}_3(\xi,\eta,\zeta)=\eta$ and $\varphi^{0}_4(\xi,\eta,\zeta)=\zeta$. (2.28) is now calculated as

$$M_{pq}=det(\mathbf{J})\int_{T_0}\varphi^0_p(\xi,\eta,\zeta) \varphi^0_q(\xi,\eta,\zeta) d\xi d\eta d\zeta,\quad p,q=1,2,3,4.$$ (2.30)

$\mathbf{J}$ is the Jacobian of the projection ( $\ref{eq:finite-element-spaces-and-meshes:5}$),

$$\mathbf{J}=\left(\begin{array}{ccc} x_1^2-x_1^1 & x_1^3-x_1^1 & ... ...& x_2^4-x_2^1 x_3^2-x_3^1 & x_3^3-x_3^1 & x_3^4-x_3^1 \end{array}\right).$$ (2.31)

Figure 2.1: Tethraedal element in $(x,y,z)$-coordinate system

Figure 2.2: Tethraedal element in normalized $(\xi ,\eta ,\zeta )$-coordinate system