Blow-up of continuous and semidiscrete solutions to elliptic equations with semilinear dynamical boundary conditions of parabolic type

In this paper we present existence of blow-up solutions for elliptic equations with semilinear boundary conditions that can be posed on all domain boundary as well as only on a part of the boundary. Systems of ordinary differential equations are obtained by semidiscretizations, using finite elements in the space variables. The necessary and sufficient conditions for blow-up in these systems are found. It is proved that the numerical blow-up times converge to the corresponding real blow-up times when the mesh size goes to zero.