Mass Matrix

The mass matrix is the Gram matrix of the basis in $L^2(\Omega )$:

$$M_{ij}={\int }_{\Omega }{\phi }_i{\phi }_j\mathrm{d}x.$$

For $u_h=\sum u_i{\phi }_i$ and $v_h=\sum v_i{\phi }_i$: $(u_h,v_h{)}_{L^2}={\mathbf{u}}^{\top }M\mathbf{v}$. So $\Vert u_h{\Vert }_{L^2}^2={\mathbf{u}}^{\top }M\mathbf{u}$. The Euclidean norm of the coefficient vector is not the $L^2$ norm of the function, except up to scaling on uniform meshes.

Properties. Symmetric positive definite and sparse, with condition number $\mathcal{O}(1)$ independent of $h$ on quasi-uniform meshes. For $P_0$ elements it is diagonal with the cell areas.

Assembly. A sum over cells of local matrices $M_e[i,j]={\sum }_q{\phi }_i(x_q){\phi }_j(x_q)w_q|\det J_e(x_q)|$ (see Numerical Quadrature and Assembly).

Uses.

• $L^2$ inner products and norms of discrete functions, for example in $L^2$-Tikhonov Regularization;
• the L2 Projection $M\mathbf{z}=\mathbf{b}$ of non-polynomial functions such as the adjoint gradient $-\nabla u\cdot \nabla \lambda$;
• turning a gradient vector (a functional, entries $\partial J/\partial z_i$) into a function: the $L^2$ Riesz representer is $M^{-1}{\nabla }_{z}J$ (see Gradient Representation and the Riesz Map).

Since $M$ is fixed for a given mesh, a sparse Cholesky factorisation is computed once and reused.

In ModularEIT.jl: assemble_mass!, FEMatrices.

References

1. S. C. Brenner, L. R. Scott (2008). The Mathematical Theory of Finite Element Methods, 3rd ed. Springer. doi:10.1007/978-0-387-75934-0
2. A. Ern, J.-L. Guermond (2004). Theory and Practice of Finite Elements. Springer. doi:10.1007/978-1-4757-4355-5