A Semirefining Extrapolation Multigrid Method With Fourth-Order Compact Schemes for 2D Nonlinear Elliptic Equations

A novel semirefining extrapolation nonlinear multigrid (SENMG) method is proposed for computing high-accuracy numerical solutions of two-dimensional (2D) nonlinear elliptic equations discretized by fourth-order compact finite difference schemes. The method starts on the coarsest grid and employs a combination of full-refining and semirefining strategies to construct a hierarchy of auxiliary grids. At each grid level, a line Gauss–Seidel nonlinear multigrid (LGSNMG) method is developed as a subsolver for the resulting discrete nonlinear systems. An optimized initial approximation is generated by integrating an extrapolation interpolation with quintic Lagrange interpolation, thereby accelerating the convergence of the LGSNMG iterations. On the finest grid, a high-precision solution is constructed by a completed Richardson extrapolation technique. Numerical experiments demonstrate that the proposed SENMG method achieves substantial efficiency gains over existing approaches when solving large-scale nonlinear systems arising from 2D nonlinear elliptic equations.