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We introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order  ⁢   (   h  +   Δ  ⁢  t    ){\mathcal{O}(h+\Delta t)}    in a certain energy norm defined precisely in the article. We only assume the realistic regularityu  t   ∈    L  2   ⁢   (  0  ,  T  ;    L  2   ⁢   (  Ω  )    ){u_{t}\in L^{2}(0,T;L^{2}(\Omega))}    and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

Crouzeix–Raviart Finite Element Approximation for the Parabolic Obstacle Problem