Assume \(\mathcal{V}_h\) is a finite element space with the nodal basis \(\left\{ \phi_1,\phi_2,\cdots,\phi_n \right\}\). Using this basis as the trial space, the continuous function \(f\) can be discretized as \(f_h\) by following the ansatz \(f_h=\sum_{i=1}^n a_i\phi_i\). There are two ways to achieve this: interpolation and projection (or more specifically, \(L_2\)-projection).

Interpolation means ensuring \(f_h\) to have exact function values as \(f\) at all support points of the trial functions. The support point of a trial function is the point where its value equals one and this is obvious for a nodal basis function \(\phi_j\), because \(l_i(\phi_j) = \delta_{ij}\) should be satisfied, where \(l_i\) is the \(i\)-th degree of freedom in the dual space of the finite element space. Let \(\left\{ x_1,x_2,\cdots,x_n \right\}\) be the list of nodal points associated with the basis functions and evaluate \(f_h\) at each point \(f_h(x_k)=\sum_{i=1}^n a_i\phi_i(x_k)=\sum_{i=1}^n a_i\delta_{ik}=a_k \quad k=1,\cdots,n.\) Then \(a_k\) can be directly assigned with the value \(f(x_k)\).

To compute the \(L_2\)-projection of \(f\) onto \(\mathcal{V}_h\), we need to ensure that for any function \(v_h\) in \(\mathcal{V}_h\), its \(L_2\) inner product with \(f_h\) and \(f\) should be the same, i.e. \(f\) is equivalent to \(f_h\) in the sense of \(L_2\)-projection. Then the following equation should be solved \((f_{h},\phi_i) =(f,\phi_i) \quad i=1,\cdots,n.\) This is equivalent to

\[\begin{equation} \left(\sum_{j=1}^n a_j\phi_j,\phi_i\right) =(f,\phi_i) \quad i=1,\cdots,n. \label{eq:projection} \end{equation}\]

In matrix form, this is \(M F_h = F\), where \(M_{ij}=\int_{\Omega}\phi_i\phi_j dx\) is the mass matrix, \(F_i=\int_{\Omega} f\phi_i dx\) and \(F_h=(a_1,\cdots,a_n)^T\) is the vector of expansion coefficients.

From above we can conclude the following points:

  1. Unless the adopted finite element nodal basis is orthonormal, i.e. \(\int_{\Omega}\phi_i\phi_j dx = \delta_{ij}\), the mass matrix \(M\) is usually not an identity matrix and the \(L_2\)-projection of \(f\) onto \(\mathcal{V}_h\) is not a simple inner product with each basis function \(\phi_i\), but the vector of inner product values \(F\) should further be left-multiplied by the inverse of the mass matrix \(M\).
  2. Interpolation and \(L_2\)-projection of a function onto a finite element space usually generate different vectors of expansion coefficients \(\{a_i\}_{i=1}^n\) for \(f_h\). Because interpolation produces accurate function values at support points of nodal basis functions, it is suitable for assigning boundary conditions or initial values to the partial differential equation.
  3. From a differential geometry point of view, the mass matrix \(M\) can be considered as a metric tensor \(g\) (2-rank covariant tensor) with respect to the function space \(\mathcal{V}_h\) with the basis \(\left\{ \phi_1,\cdots,\phi_n \right\}\). The right hand side vector \(F\) in Equation \eqref{eq:projection} can be considered as a cotangent vector belonging to the dual space. After applying the inverse of the metric tensor \(M^{-1}\), which is equivalent to the \(\sharp\) operator in differential geometry, \(F\) is transformed to the tangent vector \(F_h\).

Backlinks: 《Domain, range and dual spaces in BEM》