The discretisation of the forward problem with finite elements: from the Galerkin principle to the matrices and operators that the solvers and the reconstruction work with.
Reading order.
- The method: Galerkin Method, Lagrange Finite Elements, Numerical Quadrature and Assembly.
- The matrices: Mass Matrix, Stiffness Matrix, Weighted Stiffness Matrix, and the Conductivity Tensor that makes the dependence on σ linear and explicit.
- Boundary conditions and electrodes: Enforcing Dirichlet Conditions, Discrete Electrode Models, Boundary Mass and Stiffness Matrices, Discrete Boundary Operator, Discrete Fractional Sobolev Norms.
- The singular Neumann problem: Null Space of the Neumann Problem, Grounding of the Potential.
- Moving functions between representations: L2 Projection, Pixel Images and Finite Element Functions.
A summary of the chain from the model to the linear systems is From Physics to Linear Algebra.
Next. Solving the systems: Linear Solvers; choosing and adapting the mesh: Meshes and Geometry.
References
- 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