Galerkin Method

A Galerkin method approximates a variational problem

$$\text{find }u\in V:a(u,v)=\ell (v)\forall v\in V$$

by replacing the infinite-dimensional space $V$ with a finite-dimensional subspace $V_h=\mathrm{span}\{{\phi }_1,\dots ,{\phi }_n\}$. Writing $u_h={\sum }_j{u}_j{\phi }_j$ and testing with each ${\phi }_i$ gives the linear system

$$A\mathbf{u}=\mathbf{b},A_{ij}=a({\phi }_j,{\phi }_i),b_i=\ell ({\phi }_i).$$

For the Conductivity Equation, $a(u,v)={\int }_{\Omega }\gamma \nabla u\cdot \nabla v$, so $A$ is the Weighted Stiffness Matrix $L_{\gamma }$. With Neumann data, $\ell (v)={\int }_{\partial \Omega }gv$.

Choice of basis.

• Finite elements: piecewise polynomials on a mesh, giving sparse matrices and complex geometries (see Lagrange Finite Elements);
• spectral / Fourier / Chebyshev bases: dense but very accurate on simple domains;
• wavelets.

Céa’s lemma. If $a$ is bounded (constant $C$) and coercive (constant $\alpha$) on $V$, the Galerkin solution is quasi-optimal:

$$\Vert u-u_h{\Vert }_V\le \frac{C}{\alpha }\underset{v_h\in V_h}{\inf }\Vert u-v_h{\Vert }_V.$$

The approximation error of the space therefore controls the discretisation error. For EIT, $C/\alpha ={\gamma }_{\max }/{\gamma }_{\min }$ (see Lax-Milgram Theorem).

In an EIT reconstruction the Galerkin solve is the inner step of each iteration (see Iterative Reconstruction Loop).

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