Boundary Mass and Stiffness Matrices

Boundary data (voltages $f$, currents $g$) are functions on $\partial \Omega$. With the traces of the basis functions restricted to the boundary DOFs, define

$$(M_{\Gamma }{)}_{ij}={\int }_{\partial \Omega }{\phi }_i{\phi }_j\mathrm{d}s,(K_{\Gamma }{)}_{ij}={\int }_{\partial \Omega }{\nabla }_{\Gamma }{\phi }_i\cdot {\nabla }_{\Gamma }{\phi }_j\mathrm{d}s,$$

where ${\nabla }_{\Gamma }$ is the tangential (surface) gradient. They are assembled like the volume matrices, but by looping over boundary facets with facet quadrature (see Numerical Quadrature and Assembly).

• $M_{\Gamma }$ is the $L^2(\partial \Omega )$ Gram matrix: $\Vert f_h{\Vert }_{L^2(\partial \Omega )}^2={\mathbf{f}}^{\top }{M}_{\Gamma }\mathbf{f}$.
• $K_{\Gamma }$ is the discrete Laplace–Beltrami operator on the boundary curve or surface. It is positive semidefinite, with kernel = constants on each connected boundary component.

Why it matters. The Euclidean product ${\mathbf{a}}^{\top }\mathbf{b}$ of nodal values is not the $L^2(\partial \Omega )$ inner product. It over-weights regions with fine boundary meshes. Consistent choices are:

• Data Fidelity Terms: $\frac{1}{2}(\mathbf{u}-\mathbf{f}{)}^{\top }{M}_{\Gamma }(\mathbf{u}-\mathbf{f})$. The adjoint right-hand side is then $M_{\Gamma }(\mathbf{u}-\mathbf{f})$.
• Load vector of a current density $g$: $\mathbf{g}=M_{\Gamma }{\mathbf{g}}_{\text{nodal}}$ (interpolate, then multiply).
• Mean zero: $1^{\top }{M}_{\Gamma }\mathbf{f}=0$.
• SVD of boundary operators in the correct geometry: transform with $M_{\Gamma }^{1/2}$ (see Truncated SVD Regularization).

The pair $(K_{\Gamma },M_{\Gamma })$ also defines the Discrete Fractional Sobolev Norms.

In ModularEIT.jl: assemble_boundary_mass!, FEMatrices.

References

1. M. Arioli, D. Loghin (2009). Discrete Interpolation Norms with Applications. SIAM J. Numer. Anal. 47(4), 2924–2951. doi:10.1137/080729360
2. A. Ern, J.-L. Guermond (2004). Theory and Practice of Finite Elements. Springer. doi:10.1007/978-1-4757-4355-5