In solving discretized elliptic problems, the idea of using two-stage iterative methods (with inner ADI iterations for model systems) turned out to be very productive (theoretical results and practical applications can be found in the reviewer’s book “Optimization in solving elliptic problems”, CRC Press, Boca Raton, 1995). In contrast to this approach, the authors try to apply methods of ADI type not for specially chosen model systems but for the given nonlinear systems directly and thus agree to apply iterations with relatively slow convergence but good parallel nature.
Special attention is paid to nonlinear possibly multivalued operators but numerical examples are given only for the Dirichlet problem for the equation \(-\Delta u+ u+ e^{2+ u}= f\) and an obstacle problem with the constraint \(u\geq 0\). A comparison is made with methods of Newton type and ILU preconditioners. The authors consider their parallel algorithms as “very honorable competitors” of classical serial algorithms.