Stability and accuracy of finite element direct solvers for Electrical Impedance Tomography

Electrical impedance tomography (EIT) of closed conductive media is an ill-posed inverse problem. In the general case, the resolution of the direct problem, derived from a second order partial derivative equation, requires a numerical approximation. The solution brought by the Finite Element Method (FEM) is often used in EIT, because it preserves the nonlinear dependence of the observation set upon the conductivity distribution. This paper addresses the reliability of numerical FEM direct solvers, as basic tools in 2D EIT inversion methods. Reliability is closely related to the discretization error of the FEM model but also to its numerical stability. Finely discretized FEM models yield reduced discretization errors. Meanwhile, special attention must be paid to stability since a more accurate FEM direct model involves a magnification of input round-off errors. A theorem is established that allows easy checking of the numerical stability through the computation of the condition number of the uniform stiffness matrix of the FEM solver. When the latter is stable, global inaccuracy reduces to the discretization error of the FEM approximation. Simulations reveal that, provided input current patterns are spatially smooth, the variations of the conductivity distribution have a limited but non negligeable influence on the discretization error.