In this talk, we first develop and analyze two-grid/multi-level algorithms via mesh refinement in the abstract framework of Brezzi, Rappaz, and Raviart for approximation of branches of
nonsingular solutions. Optimal fine grid accuracy of two-grid/multi-level algorithms can be achieved via the proper scaling of relevant meshes. An important aspect of the proposed
algorithm is the use of mesh refinement in conjunction with Newton-type methods for system solution in contrast to Newton's method on a fixed mesh. 

Then, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level
algorithm on adaptive meshes retains quadratic convergence of Newton's method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established
for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear
problem.

As applications of our theory, we consider the pseudostress-velocity formulation of Navier-Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a
posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.

Reference:

1. Dongho Kim, Eun-Jae Park, Boyoon Seo, A unified framework for two-grid methods for a class of nonlinear problems, Calcolo, December 2018, 55:45

2. Dongho Kim, Eun-Jae Park, Boyoon Seo, Optimal Error Estimates for the Pseudostress Formulation of the Navier-Stokes Equations, Applied Mathematics Letters, Volume 78, April 2018, pp 24-30

3. Dongho Kim, Eun-Jae Park, Boyoon Seo,  Convergence of Multi-level Algorithms for a Class of Nonlinear Problems. J. Sci. Comput. 84 (2020), no. 2, Paper No. 34, 23 pp.

4. Dongho Kim, Eun-Jae Park, Boyoon Seo,  Adaptive Multi-level Algorithms for a Class of Nonlinear Problems,  Invited paper in Comput. Meth. Appl. Math.