A posteriori error estimates for the problem of electrostatics with a dipole source

Electroencephalography is a non-invasive technique for detecting brain activity from the measurement of the electric potential on the head surface. In mathematical terms, it reduces to an inverse problem in which the goal is to determine the source that has generated the electric field from measurements of boundary values of the electric potential. Since for reasonable models the time-variation of the electric and magnetic fields can be disregarded, the mathematical modeling of the corresponding forward problem leads to an electrostatics problem with a current dipole source. This is a singular problem, since the current dipole model involves first-order derivatives of a Dirac delta measure. Its solution lies in LpLp for 1≤p<3/21≤p<3/2 in three dimensional domains and 1≤p<21≤p<2 in the two dimensional case.We consider the numerical approximation of the forward problem by means of standard piecewise linear continuous finite elements. We prove a priori error estimates in the LpLp norm. Then, we propose a residual-type a posteriori error estimator. We prove that it is reliable and efficient; namely, it yields global upper and local lower bounds for the corresponding norms of the error. Finally, we use this estimator to guide an adaptive procedure, which is experimentally shown to lead to an optimal order of convergence.