Analysis of steady-state nonlinear problems via gradual introduction of nonlinearity

In this paper, we develop a method that allows one to consider the nonlinearity for all the nonlinear stationary partial differential equations in a gradual way, i.e., via small increments, which inspired us to define it as an incremental approach. The method considers the tensors of the equations as being constituted of two parts, one independent and the other dependent on the solution. The second part is progressively taken into account through a convenient set of parameters. Moreover, we present a scheme that allows the incremental parameters to be always improved, either to ensure that a given iteration solution is within the convergence radius or to reduce the computational cost. Numerical results are presented for various nonlinear stationary partial differential equations, including Navier-Stokes. The results demonstrate the excellent performance of the proposed method, particularly on the lid-driven cavity problem simulated with number of Reynolds equal to 50000. Finally, in the Appendix, we suggest some sets of incremental parameters for some stationary nonlinear problems.