The convergence rate of a projection-difference method for an abstract coupled problem (Q763520)

The paper is devoted to mathematical aspects arising in the study of the Cauchy problem for coupled systems of abstract operator equations in separable Hilbert spaces. The Cauchy problem of interest takes the form \[ J_1u''(t)+ L_1 u(t)+ M\theta(t)= f_1(t),\quad J_2\theta'(t)+ L_2\theta(t)+ Nu'(t)= f_2(t), \] \[ u(0)= u'(0)= \theta(0)= 0. \] A typical application is the classical coupled system of thermoelasticity, where the first equation describes the elastic behavior, and the second one the thermal behavior. A coupling condition between the operators \(M\) and \(N\) is prescribed. In a previous paper, the author proved a well-posedness result for strong solutions in a functional-analytical setting. Among other things, a priori estimates were given for the strong solution \((u,\theta)\). In the paper under consideration, the author proposes a projection-difference method for simulating the strong solution by a sequence of iterates. On the one hand, uniform a priori estimates are given for the sequence of iterates; on the other hand, an estimate for the distance of the iterates to the strong solution is proved. The approximation scheme bases on the Galerkin method for spatial variables (separable Hilbert spaces are used) and on a multi-level time-difference scheme. Several parameters appear. As usual, supposed relations between these parameters guarantee the stabilization of the scheme.