20. Finite Element Method

Ciarlet’s definition of a finite element is:

Definition: A Finite element is a triple \((T, V_{T}, \Psi_{T})\), where

• \(T\) is a bounded set

• \(V_{T}\) is function space on \(T\) of finite dimension \(N_T\)

• \(\Psi_{T} = \{ \psi^1_T, \ldots , \psi^{N_T}_T \}\) is a set of linearly independent functionals on \(V_{T}\).

Typical sets \(T\) are intervals, triangles, quadrilaterals, tetrahedra, …

The functionals \(\{ \psi^i_T \}\) are called degrees of freedom.

The nodal basis \(\{\varphi^1_T\, \ldots \varphi^{N_T}_T\}\) for \(V_T\) is the basis dual to \(\Psi_T\), i.e.,

\[ \psi^i_T (\varphi^j_T) = \delta_{ij} \]

Barycentric coordinates are useful to express the nodal basis functions.

Finite elements with point evaluation functionals are called Lagrange finite elements, elements using also derivatives are called Hermite finite elements.

Usual function spaces on \(T \subset {\mathbb R}^2\) are

\[\begin{align*} P^p & := \mbox{span} \{ x^i y^j : 0 \leq i, \, 0 \leq j, \, i+j \leq p \}, \\ Q^p & := \mbox{span} \{ x^i y^j : 0 \leq i \leq p, \, 0 \leq j \leq p \}, \end{align*}\]

where \(P^p\) are called polynomials of total order \(p\), and \(Q^p\) are called polynomials of partial order \(p\).

Examples for finite elements are

• A linear line segment

• A quadratic line segment

• A Hermite line segment

• A constant triangle

• A linear triangle

• A non-conforming triangle

• A Morley triangle

• A Raviart-Thomas triangle

The local nodal interpolation operator defined for functions \(v \in C^m(\overline T)\) is

\[ I_T v := \sum_{\alpha = 1}^{N_T} \psi^\alpha_T(v) \varphi^\alpha_T \]

It is a projection.

Two finite elements \((T,V_T, \Psi_T)\) and \((\widehat T, V_{\widehat T}, \Psi_{\widehat T})\) are called equivalent if there exists an invertible function \(\Phi\) such that

• \(T = \Phi (\widehat T)\)

• \(V_T = \{ \hat v \circ \Phi^{-1} : \hat v \in V_{\widehat T} \}\)

• \(\Psi_T = \{ \psi^T_i : V_T \rightarrow {\mathbb R} : v \rightarrow \psi^{\hat T}_i (v \circ \Phi) \}\)

Two elements are called affine equivalent, if \(\Phi\) is an affine-linear function.

Lagrangian finite elements defined above are equivalent. The Hermite elements are not equivalent.

Two finite elements are called interpolation equivalent if there holds

\[ I_T (v) \circ \Phi = I_{\widehat T} (v \circ \Phi) \]

Lemma: Equivalent elements are interpolation equivalent

The Hermite elements defined above are interpolation equivalent.

A regular triangulation \({\cal T} = \{ T_1, \ldots, T_M \}\) of a domain \(\Omega\) is the subdivision of a domain \(\Omega\) into closed triangles \(T_i\) such that \(\overline \Omega = \cup T_i\) and \(T_i \cap T_j\) is

• either empty

• or a common vertex, edge, or face of \(T_i\) and \(T_j\)

• or \(T_i = T_j\) in the case \(i = j\).

In a wider sense, a triangulation may consist of different element shapes such as segments, triangles, quadrilaterals, tetrahedra, hexhedra, prisms, pyramids.

A finite element complex \(\{ (T, V_T, \Psi_T) \}\) is a set of finite elements defined on the geometric elements of the triangulation \({\cal T}\).

It is convenient to construct finite element complexes such that all its finite elements are affine equivalent to one reference finite element \((\widehat T, \hat V_T, \hat \Psi_T)\). The transformation \(\Phi_T\) is such that \(T = \Phi_T (\widehat T)\).

Examples: reference linear line segment on \((0,1)\).

The finite element complex allows the definition of the global interpolation operator for \(C^m\)-smooth functions by

\[ I_{\cal T} v_{|T} = I_T v_T \qquad \forall \, T \in {\cal T} \]

The finite element space is

\[ V_{\cal T} := \{ v = I_{\cal T} w : w \in C^m(\overline \Omega) \} \]

We say that \(V_{\cal T}\) has regularity \(r\) if \(V_{\cal T} \subset C^r\). If \(V_{\cal T} \neq C^0\), the regularity is defined as \(-1\).

Examples:

• The \(P^1\) - triangle with vertex nodes leads to regularity \(0\).

• The \(P^1\) - triangle with edge midpoint nodes leads to regularity \(-1\).

• The \(P^0\) - triangle leads to regularity \(-1\).

For smooth functions, functionals \(\psi_{T,\alpha}\) and \(\psi_{\widetilde T, \tilde \alpha}\) sitting in the same location are equivalent. The set of global functionals \(\Psi = \{ \psi_1, \ldots, \psi_N\}\) is the linearly independent set of functionals containing all (equivalence classes of) local functionals.

The connectivity matrix \(C_T \in {\mathbb R}^{N \times N_T}\) is defined such that the local functionals are derived from the global ones by

\[ \Psi_T (u) = C_T^t \Psi (u) \]

Examples in 1D and 2D

The nodal basis for the global finite element space is the basis in \(V_{\cal T}\) dual to the global functionals \(\psi_j\), i.e.,

\[ \psi_j(\varphi_i) = \delta_{ij} \]

There holds

\[\begin{align*} \varphi_{i|_T} & = I_T \varphi_i = \sum_{\alpha = 1}^{N_T} \psi_T^\alpha (\varphi_i) \varphi^\alpha_T \\ & = \sum_{\alpha = 1}^{N_T} (C_T^t \Psi(\varphi_i))_\alpha \varphi_T^\alpha \\ & = \sum_{\alpha = 1}^{N_T} (C_T^t e_i)_\alpha \varphi_T^\alpha = \sum_{\alpha = 1}^{N_T} C_{T,i\alpha} \varphi_T^\alpha \end{align*}\]