On the discretization in time of semilinear parabolic equations with nonsmooth initial data

Source abstract

Single-step discretization methods are considered for equations of the form u t + A u = f ( t , u ) {u_t} + Au = f(t,u) , where A is a linear positive definite operator in a Hilbert space H . It is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize, even if the initial data v are only in H , but also that, in contrast to the situation in the linear homogeneous case, higher-order convergence is not possible in general without further assumptions on v .