  The focus of this paper is on the concurrent reconstruction of both the
diffusion and potential coefficients present in an elliptic/parabolic equation,
utilizing two internal measurements of the solutions. A decoupled algorithm is
constructed to sequentially recover these two parameters. In the first step, we
implement a straightforward reformulation that results in a standard problem of
identifying the diffusion coefficient. This coefficient is then numerically
recovered, with no requirement for knowledge of the potential, by utilizing an
output least-square method coupled with finite element discretization. In the
second step, the previously recovered diffusion coefficient is employed to
reconstruct the potential coefficient, applying a method similar to the first
step. Our approach is stimulated by a constructive conditional stability, and
we provide rigorous a priori error estimates in $L^2(\Omega)$ for the recovered
diffusion and potential coefficients. Our approach is stimulated by a
constructive conditional stability, and we provide rigorous a priori error
estimates in $L^2(\Omega)$ for the recovered diffusion and potential
coefficients. To derive these estimates, we develop a weighted energy argument
and suitable positivity conditions. These estimates offer a beneficial guide
for choosing regularization parameters and discretization mesh sizes, in
accordance with the noise level. Some numerical experiments are presented to
demonstrate the accuracy of the numerical scheme and support our theoretical
results.
