Lagrange Finite Elements

Lagrange elements use piecewise polynomial basis functions defined by their values at nodes. ${\phi }_i(x_j)={\delta }_{ij}$, so the coefficient vector is the vector of nodal values.

• $P_k$ on triangles/tetrahedra: complete polynomials of degree $\le k$.
• $Q_k$ on quadrilaterals/hexahedra: tensor-product polynomials of degree $\le k$ in each variable. $Q_1$ on a square uses the bilinear functions $1,x,y,xy$ and has 4 nodes.
• $P_0$ / $Q_0$: piecewise constants, one value per cell, discontinuous.

Continuity. $P_k/Q_k$ with $k\ge 1$ give globally continuous functions, a subspace of $H^1(\Omega )$ (conforming). $P_0$ is only in $L^2$.

Choices for EIT.

• The potential $u$ needs an $H^1$-conforming space: $Q_1$, $P_1$ or higher.
• The conductivity $\sigma$ can use a separate discretisation: $P_0$ (one value per cell, natural for $L^{\infty }$ and for Total Variation with jumps) or $Q_1/P_1$ (nodal, needed for the $H^1$ seminorm in Tikhonov Regularization).
• Structured $n\times n$ quadrilateral grids on $[-1,1{]}^2$ make $\sigma$ an image. Standard image processing and convolutional networks then apply directly (see Invariant and Equivariant Functions).

Error estimates. For smooth $u$ and $Q_k/P_k$ elements on a quasi-uniform mesh of size $h$: $\Vert u-u_h{\Vert }_{H^1}=\mathcal{O}(h^k)$ and $\Vert u-u_h{\Vert }_{L^2}=\mathcal{O}(h^{k+1})$. With discontinuous conductivities the solution has limited regularity, and the rates drop unless the mesh follows the discontinuities.

In ModularEIT.jl: FerriteDiscretization.

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