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Fading regularization FEM algorithms for the Cauchy problem associated with the two‐dimensional biharmonic equation

In this paper, we use the fading regularization method to solve a biharmonic inverse problem, represented by the Cauchy problem. Two formulations are studied and implemented numerically using a finite element method (FEM). We present numerical reconstructions of the missing data on the inaccessible part of the boundary from the knowledge of over‐prescribed noisy data for both smooth and piecewise smooth two‐dimensional geometries. Numerical examples validate the convergence, stability and efficiency of the proposed numerical algorithm, as well as its capability to deblur the noisy data.