Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: error estimates. (Q851564)

In the paper the second order semilinear parabolic initial boundary value problem in a bounded domain \(\Omega \subset \mathbb R^N\) is studied. On nonadjacent parts \(\Gamma _{\text{non}}\) and \(\Gamma _{\text{Dir}}\) of the boundary \(\partial \Omega \) author imposes nonlocal and Dirichlet boundary conditions. On the remaining part \(\Gamma _{\text{Neu}}\) of \(\partial \Omega \), a Robin type boundary condition is set up. For the construction of a weak solution the Rothe method of time discretization is used. The a-priori estimates obtained by the choice of suitable test functions in the weak formulation of approximated problem are then used in the proof of convergence of the scheme. In the last section of the paper the error estimates for time discretization are computed and in two examples the exact solutions are compared with the numerical approximations.