Newton-Coupling of Fixed Point Iterations

Summary

To solve a coupled system of two equations it may be intended not to use the Newton-Raphson method, for example due to the non-sparsity of the Jacobian of the entire system or because there exist solvers for the subsystems. For this type of problems we present an iterative Newton type method which requires only iterative solution steps for the single equations. The algorithm is based on a formal Block-Gauss elimination of the full Newton system and the solution of the resulting Schur complement equation by a Bi-CGSTAB iteration. No computation of the Jacobian of the whole system is necessary. Efficient alternative approaches demand that one of the incorporated systems has relatively small dimension. On the contrary our approach allows similar sizes of the subsystems. This is needed for the intended application which is a system of PDEs describing the behaviour of a chemical reactor for the combustion of coal. Our numerical examples deal with this combustion model.