To solve equation (7) numerically, we first applied a Galerkin formulation and then utilized the finite element method with linear basis functions and tetrahedral volume elements. The finite element method has three strengths over other methods for this type of computation. First, it allows explicit modeling of the anisotropic conductivities of skeletal muscle. The finite element method computes an estimate of the potential field over each element, taking into account the material properties of each individual element. Therefore, it is possible to specify different conductivity tensors over different regions, or even for each element. Secondly, the finite element method supports unstructured meshes. This permits us to use an adaptive, unstructured mesh density that can model complicated geometries while simultaneously minimizing numerical discretization error. Finally, as long as tissue regions are identified and boundaries are maintained, the finite element method allows very flexible assignment of tissue characteristics. For example, altering the conductivity of the lungs required changing only a single entry in a look-up table that linked tissue types with conductivity values.
Applying the finite element method to equation (7) results in a system of linear equations, , where the matrix A contains the geometry and conductivity information for the problem and has the properties of being sparse, symmetric, and positive definite. The solution of this system provides the potential at each node in the discretized model geometry. Because the matrix A is symmetric and positive definite, a Choleski preconditioner can be used to accelerate the rate of convergence of iterative solvers. We utilized a preconditioned Incomplete Choleski Conjugate Gradient (ICCG) method to iteratively solve the linear system.