Semilinear Parabolic Equations

In the last chapter we considered discretization in both space and time of a model nonlinear parabolic equation. The discretization with respect to space was done by piecewise linear finite elements and in time we applied the backward Euler and Crank-Nicolson methods. In this chapter we shall restrict the consideration to the case when only the forcing term is nonlinear, but discuss more general approximations in the spatial variable. We shall begin with the spatially semidiscrete problem and first briefly study global conditions on the forcing term and the finite element spaces under which optimal order error estimates can be derived for smooth data, uniformly down to t = 0, and then turn our attention to the analysis for nonsmooth initial data. We then discuss discretization in time by the backward Euler method, in particular with reference to nonsmooth initial data.