Robust multigrid technique for solving partial differential equations on structured grids

Abstract

A new robust multigrid technique for solving elliptic partial differential equations is proposed. The technique is based on a united computational algorithm that consists of the following stages: 1) adaption of equations to numerical methods, 2) the control volume discretization, and 3) applying multigrid iterations. Special subgrids of the finest grid are generated to obtain the most powerful coarse grid correction strategy. Accuracy of the transfer operators is independent of the mesh size on coarse grids; therefore, a smoothing procedure and a multigrid cycle may be very simple. Expanded robustness of the multigrid technique is a result of adaption of equations, extremely accurate formulation of the discrete problems on the coarse grids, original coarsening, the most powerful coarse grid correction strategy, construction of problem-independent transfer operators, and absence of pre-smoothing and interpolation. The paper represents the algorithm, estimates of computational work, and results of numerical tests performed. Our numerical tests demonstrate robustness and efficiency of our multigrid technique.