Weighted Stiffness Matrix

The system matrix of the discretised Conductivity Equation is the conductivity-weighted stiffness matrix

$$(L_{\gamma }{)}_{ij}={\int }_{\Omega }\gamma (x)\nabla {\phi }_i\cdot \nabla {\phi }_j\mathrm{d}x.$$

With Neumann data $g$, the discrete forward problem is $L_{\gamma }\mathbf{u}=\mathbf{g}$ with the load vector ${\mathbf{g}}_i={\int }_{\partial \Omega }g{\phi }_i\mathrm{d}s$.

Assembly. Loop over cells and interpolate $\gamma$ at the quadrature points $x_q$. The conductivity may live on its own mesh or space, for example $P_0$, or $Q_1$ on the same grid:

$$L_e[i,j]=\sum _q\gamma (x_q)\nabla {\phi }_i(x_q)\cdot \nabla {\phi }_j(x_q)w_q|\det J_e(x_q)|,$$

then scatter into the global sparse matrix (see Numerical Quadrature and Assembly).

Properties.

• Symmetric positive semidefinite with kernel = constants, like the Stiffness Matrix.
• Spectrally equivalent to $K$: ${\gamma }_{\min }K\le L_{\gamma }\le {\gamma }_{\max }K$.
• Linear in $\gamma$: $L_{\gamma }={\sum }_k{\gamma }_k{L}^{(k)}$ for a basis expansion $\gamma ={\sum }_k{\gamma }_k{\psi }_k$. The solution $\mathbf{u}=L_{\gamma }^{-1}\mathbf{g}$ is nonlinear in $\gamma$ nonetheless (see Forward Map).
• Its sparsity pattern does not depend on $\gamma$, so the symbolic structure and the preconditioner setup can be reused across iterations. Together with linearity this makes reassembly a single sparse matrix-vector product (see Conductivity Tensor).

Every reconstruction iteration reassembles $L_{\sigma }$ for the current guess $\sigma$ and then solves the state and adjoint systems with it (see Adjoint State Method).

In ModularEIT.jl: assemble_weighted_stiffness!, ConductivityTensor.

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. Adler, W. R. B. Lionheart (2006). Uses and abuses of EIDORS: an extensible software base for EIT. Physiol. Meas. 27(5), S25–S42. doi:10.1088/0967-3334/27/5/S03