The multigrid methods with local mesh refinement provide one solution method to achieve efficient local resolution by solving problems on various local nested grids, and by using these grids as a basis for fast solution and correction on the global basic grid of the calculation domain.
In this study, we propose to focus on the comparison of three methods based on this concept of hierarchical multigrid local refinement: L.D.C. - Local Defect Correction - (Hackbush, 1984), F.A.C. - Fast Adaptive Composite - (McCormick, 1986), and the F.I.C. - Flux Interface Correction - algorithm that we proposed recently. These methods are tested on two examples of a bidimensional elliptic problem, presenting respectively discontinuities of the operator coefficients and singularities of the exact solution. We compare, for V-cycle procedures, the asymptotic evolution of the corresponding local and global (with discrete norms) errors, and the convergence rate of these algorithms.
At last, we present an original algorithm based on the coupling of the FIC method with a divergence free Navier-Stokes solver in the primitive variables formulation. First results will be discussed for Poiseuille or Stokes flows and Navier-Stokes flows around a circular cylinder or inside a lid-driven cavity.
Ph. ANGOT, J.P. CALTAGIRONE, K. KHADRA, An Adaptive Method for Local Mesh Refinement: the Flux Interface Correction, C.R. Acad. Sci. Paris, t. 315, Série I, p. 739-745, 1992.