Analysis of discretizations of parabolic problems in pairs of spaces (Q1569879)

Single step discrete approximations to an abstract Cauchy problem are studied. Let \((E,V)\) be a pair of Banach spaces, where \(E\) is densely imbedded in \(V\), and let \(A\) be some (generally unbounded) operator in \(E\). Given \(u_0\in V\) consider the initial value problem (IVP), \(u_t+ Au=0\), \(t>0\), \(u(0)= u_0\). Associated with this IVP is the discrete problem \(U_{n+1}= r(- kA)U_n\), \(U_0= \omega(-kA)u_0\), where \(r\) and \(\omega\) are rational functions such that the operators \(r(-kA): E\to E\) and \(\omega(-kA): V\to E\) are well defined and bounded for any \(k> 0\). Two types of results are established in the paper. First, stability results of the form \(\|U_n\|_E\leq Ct^{-\gamma}_{n+1}\|u_0\|_V\), where \(t_n= kn\), \(\gamma>0\). Second, error estimates of the form \(\|U_n- u(t_n)\|_E\leq Ck^\mu t^{-\gamma-\mu}_n\|u_0\|_V\) are obtained. The case when \(A\) is the Laplace operator, \(A(u)= -\sum_i\partial^2 u/\partial x^2_i\), and the boundary conditions are of Dirichlet type is considered in particular.