A least squares finite element method for backward parabolic problems

Backward parabolic equations, such as the backward heat equation, are classical examples of ill-posed problems where solutions may not exist or depend continuously on the data. In this work, we study a least squares finite element method to numerically approximate solutions to such problems. We derive conditional stability estimates for the weak formulation of inhomogeneous backward parabolic equations, assuming minimal regularity of the solution. These stability results are then used to establish \emph{a priori} error bounds for our proposed method. We address key computational aspects, including the treatment of dual norms through the construction of suitable test spaces, and iterative solutions. Numerical experiments are used to validate our theoretical findings.