Solving upwind-biased discretizations II: Multigrid solver using semicoarsening

This paper studies a novel multigrid approach to the solution for a second order upwind biased discretization of the convection equation in two dimensions. This approach is based on semicoarsening and well balanced explicit correction terms added to coarse-grid operators to maintain on coarse grids the same cross-characteristic interaction as on the target (fine) grid. Colored relaxation schemes are used on all the levels allowing a very efficient parallel implementation. The results of the numerical tests can be summarized as follows: 1) The residual asymptotic convergence rate of the proposed $V(0, 2)$ multigrid cycle is about 3 per cycle. This convergence rate far surpasses the theoretical limit ($4/3$) predicted for standard multigrid algorithms using full coarsening. The reported efficiency does not deteriorate with increasing the cycle depth (number of levels) and/or refining the target-grid mesh spacing. 2) The full multigrid algorithm (FMG) with two $V(0, 2)$ cycles on the target grid and just one $V(0, 2)$ cycle on all the coarse grids always provides an approximate solution with the algebraic error less than the discretization error. Estimates of the total work in the FMG algorithm are ranged between 18 and 30 minimal work units (depending on the target discretization). Thus, the overall efficiency of the FMG solver closely approaches (if does not achieve) the goal of the textbook multigrid efficiency. 3) A novel adaptive multigrid approach to deriving a discrete solution approximating the true continuous solution with a relative accuracy given in advance is developed. The computational complexity of this method is (nearly) optimal (comparable with the complexity of the FMG algorithm applied to solve the problem on the optimally spaced target grid).