Abstract
We consider a two-dimensional homogeneous parabolic Dirichletproblem with an obstacle inside the domain \(\Omega\times(0,T]\) .The equation contains a fractional time derivative of order \(\alpha\in(0,1)\) and a uniformly monotone quasilinear diffusionoperator. The problem is approximated by an implicit grid schemeusing \(L1\) -approximation in time and \(P_{1}\) finite elementapproximation for the diffusion operator. We prove the existenceof a unique solution to the grid scheme and establish a prioriestimates in the grid analogues of the space \(L_{2}((0,T);H^{1}(\Omega))\) for the solution and the space \(L_{2}((0,T);H^{-1}(\Omega))\) for the right-hand side. The main result of thepaper is an estimate of the closeness of the grid solution to theinterpolant of the exact solution under natural assumptionsregarding the smoothness of the exact solution and the geometry ofthe coincidence set. Namely, the accuracy estimate of order \(O(\tau^{3/2-\alpha}+h^{1/2})\) in the grid analogue of \(L_{2}((0,T);H^{1}(\Omega))\) norm is derived under the assumption of smoothnessof the exact solution at points that do not coincide with thepoints of the piecewise smooth free boundary, where adiscontinuity of the first derivatives is allowed. The results ofnumerical tests of the accuracy of the grid scheme on a sequenceof grids are presented.
