preconditioned global gpbicg method for solving saddle point problems with multiple right-hand sides and its convergence analysis

We propose the preconditioned global generalized product-type method based on the preconditioned global bicg method to solve nonsymmetric saddle point problems with multiple right-hand sides. we apply an indefinite preconditioner to enhance the convergence rate of the method. we also present some theoretical analysis and discuss the convergence of the pgl-gpbicg method. some useful properties of the preconditioned matrix are established. moreover, we present the bounds for the residual norm of the pgl-gpbicg method according to the residual norm of the global gmres method that guarantees convergence. finally, some numerical examples are presented to show the efficiency of the new method in comparison with the preconditioned global bicgstab method, and a comparison with another preconditioner is also provided.