Quantitative Analysis of the Reciprocity Gap Dichotomy for Inclusions with Variable Conductivity

We study the quantitative structure of the reciprocity gap method for inclusions with spatially varying conductivity. Motivated by the variable-coefficient reciprocity gap identity, we investigate the discrete approximation mechanism underlying the bounded/blow-up dichotomy for sampling points inside and outside the inclusion. The reciprocity gap functional is discretized by harmonic test functions, and the resulting ill-conditioned linear system is regularized by a Tikhonov term consistent with the L2(∂D) trace norm appearing in the weighted formulation. The regularization parameter is selected by the L-curve criterion. For constant, radially varying, and angularly oscillating contrasts, the numerical results show that exterior sampling points exhibit an approximately exponential growth of ∥vN(z)∥L2(∂D) with respect to the harmonic order N, whereas interior points remain bounded. This behavior is quantified through fitted growth rates and contrast indicators, and its dependence on geometry and model parameters is examined. The results provide a quantitative description of the reciprocity gap approximation mechanism in heterogeneous media and show that the bounded/blow-up dichotomy remains numerically detectable beyond the constant-coefficient setting.