On the Computation of Blow-Up Solutions of Elliptic Equations with Semilinear Dynamical Boundary Conditions

Abstract

In this paper we study numerically blow-up solutions of elliptic equations with nonlinear dynamical boundary conditions. First, we formulate a result for blow-up, when dynamical boundary condition is posed on the part of the boundary. Next, by semidiscretization, we obtain a system of ordinary differential equations (ODEs), the solution of which also blows up. Under certain assumptions we prove that the numerical blow-up time converges to the corresponding real blow-up time, when the mesh size goes to zero. We investigate numerically the blow-up set (BUS) and the blow-up rate. Numerical experiments with local mesh refinement technique are also discussed.