Numerical Reconstruction of Coefficients in Elliptic Equations Using Continuous Data Assimilation

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