One-dimensional finite element
In this chapter, we provide strict mathematical definitions related to the classical finite element method. In particular, finite element definitions were given by Ciarlet in 1978, and in the context of the adaptive finite element method by Demkowicz in 2007 [1], [2].
Both of these definitions are provided below. Ciarlet's definition uses the concept of degrees of freedom, while Demkowicz's definition uses the concept of the shape function and projection-based interpolation operator. We will then show the relationship between these two definitions that show that they are equivalent. First, we define a finite element in one dimension, for a reference (often also called a standard) finite element defined on the interval \( \hat{K}=[0,1] \subset {\cal R} \), and then we will generalize this definition to any element defined in any interval \( K=[ x_l,x_r ] \subset {\cal R} \).
\( \left( \hat{K}, X\left(\hat{K}\right), \Pi_p \right) \)
defined by the following four steps
- Geometry: \( \hat{K}=[0,1] \subset {\cal R} \)
- Selection of nodes: \( \hat{a}_1, \hat{a}_2 \) nodes associated with vertices 0 and 1 of the element, and \( \hat{a}_3 \) node associated with the interior (0,1) of the element
- Definition of element shape function
\( X \left( \hat{K}\right)=span \{ \hat{\chi}_j \in {\cal P}^p\left(\hat{K}\right),j=1,...,p+1 \} \) where \( {\cal P}^p\left(\hat{K}\right) \) are degree polynomials \( p \) specified on the interval \( \hat{K} =(0,1) \) and \( \hat{\chi}_1(\xi)=1-\xi \), \( \hat{\chi}_2(\xi)=\xi \), \( \hat{\chi}_3(\xi)=(1-\xi)\xi \), \( \hat{\chi}_l(\xi)=(1-\xi)\xi(2\xi-1)^{l-3} \quad l=4,...,p+1 \).
- Definition of the projection-based interpolation operator \( \Pi_p:H^1\left( \hat{K} \right) \rightarrow X\left( \hat{K}\right) \). For a given function \( u \in H^1\left(\hat{K} \right) \), its projection based interpolant is \( \Pi_pu\in X\left( \hat{K}\right) \) is defined by the following conditions:
\( \Pi_p u(\hat{a}_1)=u(\hat{a}_1) \)
\( \Pi_p u(\hat{a}_2)=u(\hat{a}_2) \)
\( \| \Pi_p u -u \|_{H^1_0(0,1)}\rightarrow min \)
where \( \| \Pi_p u -u \|_{H^1_0(0,1)} = \int_0^1 \left( \left( \Pi_p u \right)' -u' \right)^2 d\xi \) this is the norm in the Sobolev space \( H^1_0(0,1) \).
\( \Pi_p u(0)=u(0) \)
\( \Pi_p u(1)=u(1) \)
\( \| \Pi_p u -u \|_{H^1_0(\hat{a}_3)}\rightarrow min \)
is equivalent to the problem: find \( \Pi_pu\in X\left( \hat{K}\right) \) so that
\( \Pi_p u(0)=u(0) \)
\( \Pi_p u(1)=u(1) \)
\( \int_0^1 \left( \left( \Pi_p u \right)' -u' \right) v' d\xi =0 \quad \forall v \in X\left( \hat{K} \right) \)
which of the following is equivalent to the problem of solving the system of equations:
\( \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ a_{31} & a_{32} & a_{33} & \cdots & a_{3,p+1} \\ \vdots & \vdots & \vdots & & \vdots \\ a_{p+1,1} & a_{p+1,2} & a_{p+1,3} & \cdots & a_{p+1,p+1}\end{bmatrix} \begin{bmatrix} u^p_1 \\ u^p_2 \\ u^p_3 \\ \vdots \\ u^p_{p+1} \end{bmatrix} = \begin{bmatrix} u(0) \\ u(1) \\ b_3 \\ \vdots \\ b_{p+1} \end{bmatrix} \)
where \( \Pi_p u = \sum_{i=1,...,p+1} u^p_i \chi_i \) is a linear combination of basis functions,
\( a_{ij} = \int_0^1 \hat{\chi}_i' \hat{\chi}_j' d\xi \)
\( \left( \hat{K}, V^*\left(\hat{K}\right), X\left(\hat{K}\right) \right) \)
defined by the following four steps
- Geometry: \( \hat{K}=\left(0,1\right) \in {\cal R} \)
- Selection of nodes: \( \hat{a}_1, \hat{a}_2 \) nodes associated with the nodes 0 and 1 of the element, and \( \hat {a}_3 \) node associated with the interior (0,1) of the element
- The definition of the space of degrees of freedom \( V^* \left( \hat{K}\right) = span \{ \psi_i \}_{i=1,...,p+1} \) as a dual space to \( V\left( \hat{K} \right) \). We associate the degrees of freedom with the element nodes and with the interior of the element \( \psi_1 : V\left(\hat{K}\right) \ni f \rightarrow f(\hat{a}_1)\in R \), \( \psi_2 : V\left(\hat{K}\right) \ni f \rightarrow f(\hat{a}_2)\in R \), \( \psi_3 : V\left(\hat{K}\right) \ni f \rightarrow 3 \int_{\hat{a}_3} f'(\xi) (1-2\xi)d\xi \in R \). It is possible to extend this definition to further degrees of freedom (here we give a definition for \( p=2 \).
- Construction of an approximation space \( X\left(\hat{K}\right) \subset V\left(\hat{K}\right) \). The approximation space is spanned by the base of polynomials being a dual base to the base of the degrees of freedom \( \psi_i\left(\chi_j\right) = \delta_{ij} \)
\( \chi_j, j=1,2,...,p+1 \) denote shape functions according to definition 1, while
\( \psi_i,i=1,2,...,p+1 \) for \( p = 2 \) means degrees of freedom according to definition 2, then
\( \psi_i \left( \chi_j \right) = \delta_{ij} \)
\( \Pi_p : H^1(0,1)\rightarrow {\cal P}^p \left(\hat{K }\right) \) be an interpolation operator according to Definition 1.
Let \( \chi_j, j=1,2,...,p+1 \) stand for shape functions as defined in definition 1. Then, there is a unique set of linear and continuous functionals \( \psi_j : H^1\left(0,1\right) \rightarrow {\cal P }^p \left(0,1 \right) \) such that
defined by the following four steps
- Geometry: \( K=[ x_l,x_r] \subset {\cal R}, x_l,x_r \in {\cal R} \)
- Selection of nodes: \( a_1, a_2 \) nodes associated with vertices \( x_l,x_r \) of the element, and \( a_3 \) knot associated with the interior \( K=[ x_l,x_r] {\cal R} \) of the element
- Definition of the shape function of an element \( X \left( K\right)=\{ \chi = \hat{\chi} \cdot x_K^{-1}, \hat{\chi} \in X\left(\hat{K}\right) \} \) where \( x_K:\hat{K} \rightarrow K \) is the mapping from the pattern element \( K=[0,1] \) on the element \( K=[x_l,x_r] \) data \( \hat{K} \ni \xi \rightarrow x_K\left(\xi\right)=x_l+(x_r-x_l)=x\in K \)
- Definition of the projection-based interpolation operator \( \Pi_p:H^1\left( \hat{K} \right) \rightarrow X\left( \hat{K}\right) \) defined analogously to Definition 1.
Reference elements are used to integrate the weak formulations.
The reference elements are regular in shape.
The reference element is mapped to the given element, and its geometry is described by the map mapping from the reference element to the given element. In the Jacobian integrals of this mapping, it represents the scaling of the area (volume) of the reference element for the given element.