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Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: error estimates. (English) Zbl 1099.65081
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.

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K55 Nonlinear parabolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
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