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4.4 The Boundary Element Method

For bioelectric field problems with isotropic domains (and few inhomogeneities), another, somewhat less familiar technique, called the boundary element method, may be utilized. This technique utilizes information only upon the boundaries of interest, and thus reduces the dimension of any field problem by one. For differential operators, the response at any given point to sources and boundary conditions depends only on the response at neighboring points. The FD and FE, methods approximate differential operators are defined on subregions (volume elements) in the domain; hence, direct mutual influence (connectivity) exists only between neighboring elements, and the coefficient matrices generated by these methods have relatively few non-zero coefficients in any given matrix row. As is demonstrated by Maxwell's laws [35], equations in differential forms can often be replaced by equations in integral forms; e.g. the potential distribution in a domain is uniquely defined by the volume sources and the potential and current density on the boundary. The boundary element method utilizes this fact by transforming the differential operator defined in the domain to integral operators defined on the boundary. In the boundary element method [36,37,38], only the boundary is discretized; hence, the mesh generation is considerably simpler for this method than for the volume methods. Boundary solutions are obtained directly by solving the set of linear equations; however, potentials and gradients in the domain can be evaluated only after the boundary solutions have been obtained.

(See project in section 9).