We consider the numerical reconstruction of the spatially dependent conductivity coefficient and the source term in elliptic partial differential equations in a two-dimensional convex polygonal domain, with the homogeneous Dirichlet boundary condition and given interior observations of the solution. Using data assimilation, we derive approximated gradients of the error functional to update the reconstructed coefficients. New $L^2$ error estimates are provided for the spatially discretized reconstructions. Numerical examples are given to illustrate the effectiveness of the method and demonstrate the error estimates. The numerical results also show that the reconstruction is very robust to the errors in specific inputted coefficients.

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