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This paper concerns the study of the numerical approximation for the following initial-boundary value problem:

where  is a  convex, nondecreasing function,  is a fixed point in the domain,  and  is a positive parameter. Under some assumptions, we prove that the solution of a discrete form of the above problem blows up in a finite time and estimate its numerical blow-up time. We also show that the numerical blow-up time in certain cases converges to the real one when the mesh size tends to zero. Finally, we give some numerical experiments to illustrate our analysis.

discretization, localized semilinear parabolic equation, numerical blow-up time, convergence.