# Reference for mass matrix assembly

I will now explain what I understand to be the process of a finite element mass matrix assembly. I would like a reference which does something similar, or if I am mistaken about the process please let me know.

Let $$\Omega$$ be a bounded domain and let $$\lbrace \Omega^{\left(e\right)}\rbrace_{e=1}^{E}$$ be disjoint subsets of $$\Omega$$ which comprise a triangulation of $$\Omega$$. Let $$\mathcal{V}$$ be a finite element space over $$\Omega$$, with basis elements $$\lbrace \phi_{i}\rbrace_{i=1}^{n}$$. Let $$M\in\mathbb{R}^{n\times n}$$ be the mass-matrix

\begin{align} M_{i,j}=\sum_{e=1}^{E}\int_{\Omega^{e}}\phi_{i}\left(x\right)\phi_{j}\left(x\right)\mathrm{d}x. \end{align}

We then map the triangle $$\Omega^{\left(e\right)}$$ onto a fixed reference element $$\Omega^{r}$$ by a change of independent variable, achieved by a linear transformation.

\begin{align} M_{i,j}=\sum_{e=1}^{E}\int_{\Omega^{r}}\tilde{\phi}_{i}\left(x\right)\tilde{\phi}_{j}\left(x\right)|J^{\left(e\right)}|\mathrm{d}x. \end{align}

Because the transformation is linear the determinant of the element Jacobian $$|J^{\left(e\right)}|$$ is constant.

\begin{align} M_{i,j}=\sum_{e=1}^{E}|J^{\left(e\right)}|\int_{\Omega^{r}}\tilde{\phi}_{i}\left(x\right)\tilde{\phi}_{j}\left(x\right)\mathrm{d}x. \end{align}

Each element of the basis for $$\mathcal{V}$$ is assumed to be supported on a few elements $$\Omega^{\left(e\right)}$$, let $$m$$ be the number of basis functions with support on $$\Omega^{\left(e\right)}$$. $$m$$ is assumed to be independent of the element $$\Omega^{\left(e\right)}$$.

\begin{align} M_{i,j}=\sum_{e=1}^{E}|J^{\left(e\right)}|\int_{\Omega^{r}} \sum_{\ell,p=1}^{m} \delta_{i,e_{\ell}}\,\tilde{\phi}_{i}\left(x\right)\tilde{\phi}_{j}\left(x\right) \, \delta_{e_{p},j}\, \mathrm{d}x. \end{align}

where $$\delta$$ is the delta-Kronecker symbol and $$\lbrace e_{p}\rbrace_{p=1}^{m}$$ are the global indicies associated to a finite element function with support on $$\Omega^{\left(e\right)}$$. Let $$\lbrace \psi_{i}\rbrace_{i=1}^{m}\subset \lbrace \tilde{\phi}_{i}\rbrace_{i=1}^{n}$$ be the transformed basis functions that are supported on the reference element, those functions that are supported are element independent.

\begin{align} M_{i,j}=\sum_{e=1}^{E}|J^{\left(e\right)}|\int_{\Omega^{r}} \sum_{\ell,p=1}^{m} \delta_{i,e_{\ell}}\,\psi_{\ell}\left(x\right)\psi_{p}\left(x\right) \, \delta_{e_{p},j}\, \mathrm{d}x. \end{align}

\begin{align} M_{i,j}=\sum_{e=1}^{E}|J^{\left(e\right)}| \sum_{\ell,p=1}^{m} \delta_{i,e_{\ell}} \int_{\Omega^{r}} \psi_{\ell}\left(x\right)\psi_{p}\left(x\right) \mathrm{d}x\,\delta_{e_{p},j}. \end{align}

The only thing left to do is quadrature, so for that purpose let $$\lbrace x_{k}\rbrace_{k=1}^{q}\subset\Omega^{r}$$ be quadrature points and $$\lbrace w_{k}\rbrace_{k=1}^{q}\subset\mathbb{R}$$ be the quadrature weights.

\begin{align} M_{i,j}\approx\sum_{e=1}^{E}|J^{\left(e\right)}| \sum_{\ell,p=1}^{m} \delta_{i,e_{\ell}} \sum_{k=1}^{q} \psi_{\ell}\left(x_{k}\right)\,w_{k}\,\psi_{p}\left(x_{k}\right) \,\delta_{e_{p},j}. \end{align}

Let $$G^{\left(e\right)}\in\mathbb{R}^{m\times n}$$ be the global to local element matrix defined by $$\left(G^{\left(e\right)}\right)_{p,j}=\delta_{e_{p},j}$$, $$B\in\mathbb{R}^{q\times m}$$ be the local dof to basis functoin evaluation at the quadrature points $$B_{k,p}=\psi_{p}\left(x_{k}\right)$$ and $$D^{\left(e\right)}\in\mathbb{R}^{q\times q}$$ be a diagonal matrix which holds the quadrature weights $$D^{\left(e\right)}_{i,j}=|J^{\left(e\right)}|\delta_{i,j}w_{i}$$. With this we have

\begin{align} M\approx\sum_{e=1}^{E}\left(G^{\left(e\right)}\right)^{\top}\,B^{\top}\,D^{\left(e\right)}\,B\,G^{\left(e\right)} \end{align}

Again I am looking for textbooks which also describe this procedure or any feedback if I have any errors thanks!

In your write-up, you loop over all $$i$$ and $$j$$ and then ask which cells you need to consider to compute a particular matrix entry $$M_{ij}$$. This is expensive because most matrix entries are zero, and because you would have to visit the same cell multiple times.
• I see that looping over each $i,j$ can be expensive and one should take advantage of the sparsity to reduce the cost. Jun 2, 2021 at 20:58
• The final expression for the mass matrix seems to be what you are suggesting, that is there is a loop over elements/cells, the action of $G^{\left(e\right)}$ extracts those elements that live on the cell, etc. Jun 2, 2021 at 20:59