Inspired by Galerkin projection methods for matrix equations as studied for example by B. Beckermann [SIAM J. Numer. Anal. 49, No. 6, 2430–2450 (2011; Zbl 1244.65057)], the present paper generalizes this to Galerkin projection on tensor products of \(d\) subspaces (\(d=2\) for the Sylvester equation). This is to solve large systems with tensor product structure such as they appear in the solution of \(d\)-dimenional partial differential equations (cf. [D. Kressner and C. Tobler, SIAM J. Matrix Anal. Appl. 31, No. 4, 1688–1714 (2010; Zbl 1208.65044)]).
By exploiting the structure, it is possible to decompose the residual as a sum of \(d\) orthogonal components. Hence it only requires the computation of \(d\) simple 1D projections. Error estimates for (rational) Krylov methods extend the bounds given in Beckermann’s previously mentioned paper [loc. cit.] and do not need a multivariate analysis as in the paper by Kressner and Tobler [loc. cit.]. The convergence analysis clarifies the role of the shifts and optimal values of the shifts can be selected to improve the performance. Several numerical examples illustrate the method.