  Convergence failure and slow convergence rates are among the biggest
challenges with solving the system of non-linear equations numerically.
Although mitigated, such issues still linger when using strictly small time
steps and unconditionally stable fully implicit schemes. The price that comes
with restricting time steps to small scales is the enormous computational load,
especially in large-scale models. To address this problem, we introduce a
sequential local mesh refinement framework to optimize convergence rate and
prevent convergence failure, while not restricting the whole system to small
time steps, thus improving computational efficiency. We test the algorithm with
the non-linear two-phase flow model. Starting by solving the problem on the
coarsest space-time mesh, the algorithm refines the domain in time at water
saturation front to prevent convergence failure. It then deploys fine spatial
grid in regions with large saturation variation to assure solution accuracy.
After each refinement, the solution from the previous mesh is used to estimate
the initial guess of unknowns on the current mesh for faster convergence.
Numerical results are presented to confirm accuracy of our algorithm as
compared to the uniformly fine time step and fine spatial discretization
solution. We observe approximately 25 times speedup in the solution time by
using our algorithm.
