Stiffness Matrix

The (unweighted) stiffness matrix discretises the Laplacian:

$$K_{ij}={\int }_{\Omega }\nabla {\phi }_i\cdot \nabla {\phi }_j\mathrm{d}x,{\mathbf{z}}^{\top }K\mathbf{z}=\Vert \nabla z_h{\Vert }_{L^2(\Omega )}^2.$$

Properties.

• Symmetric positive semidefinite. Its kernel consists of the constants, since $\nabla 1=0$ and ${\sum }_i{\phi }_i\equiv 1$ for Lagrange elements. It is singular without Dirichlet conditions (see Null Space of the Neumann Problem).
• Sparse: $K_{ij}\ne 0$ only if ${\phi }_i$ and ${\phi }_j$ share a cell.
• Condition number $\mathcal{O}(h^{-2})$ on the complement of the kernel, so iterative solvers need preconditioning (see Algebraic Multigrid).
• Requires $H^1$-conforming elements ($P_1/Q_1$ or higher). For $P_0$ the gradient is zero inside cells.

Uses. $H^1$-seminorm Tikhonov Regularization ($\frac{\beta }{2}{\mathbf{z}}^{\top }K\mathbf{z}$), Laplacian smoothing, and as a Levenberg–Marquardt matrix $L_{\text{LM}}$ (see Levenberg-Marquardt Method). The conductivity-weighted version is the Weighted Stiffness Matrix.

In ModularEIT.jl: assemble_stiffness!, 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