Approximation of a mixed boundary value problem

The mixed boundary value problem for the divergent-type elliptic equation with variable coefficients is considered. It is assumed that the integration domain has a sufficiently smooth boundary that is the union of two disjoint pieces. The Dirichlet boundary condition is given on the first piece, and the Neumann boundary condition is given on the other one. So the problem has discontinuous boundary condition. Such problems with mixed boundary conditions are the most common in practice when modeling processes and are of considerable interest in the development of methods for their solution. In particular, a number of problems in the theory of elasticity, theory of diffusion, filtration, geophysics, a number of problems of optimization in electro-heat and mass transfer in complex multielectrode electrochemical systems are reduced to the boundary value problems of this type. In this paper, we propose an approximation of the original mixed boundary value problem by the third boundary value problem with a parameter. The convergence of the proposed approximations is investigated. Estimates of the approximations’ convergence rate in Sobolev norms are established.