Conforming and nonconforming finite element methods for biharmonic inverse source problem

This paper deals with the numerical approximation of the biharmonic inverse source problem in an abstract setting in which the measurement data is finite-dimensional. This unified framework in particular covers the conforming and nonconforming finite element methods (FEMs). The inverse problem is analysed through the forward problem. Error estimate for the forward solution is derived in an abstract set-up that applies to conforming and Morley nonconforming FEMs. Since the inverse problem is ill-posed, Tikhonov regularization is considered to obtain a stable approximate solution. Error estimate is established for the regularized solution for different regularization schemes. Numerical results that confirm the theoretical results are also presented.