Multigrid method for elliptic equations with anisotropic discontinuous coefficients

On Development of Parallel Algorithms for Solving Parabolic and Elliptic Equations

“…The algorithm of geometric multigrid method differs from the AMM by the presence of a sequence of roughening grids and active usage of grid constructions. The iterating operator of the multigrid method has the form (20), but the operator A H is constructed by rediscretization procedure, i.e., the approximation of the initial equation with homogeneous boundary conditions on the H-network is stored. The projection of the discrepancy r h = g h − A h u h onto the H-grid by the operator R is taken as the right-hand side of the system on the H-grid.…”

“…The projection of the discrepancy r h = g h − A h u h onto the H-grid by the operator R is taken as the right-hand side of the system on the H-grid. Algorithmic constructions of inter-grid transition operators P and R = P * are indicated in [19,20], where, in addition to the trilinear interpolation operator, the interpolation operator P based on the approximate solution of local boundary value problems is also given; this is critical for the case of discontinuous coefficients. Such inter-grid transition operators P and R = P * are said to be called problem-dependent (see [14]).…”

“…The smoothing operator S p in (20) performs the transition from the iterative approximation v to a "smoother" approximation v. Let a grid function u be an unknown exact solution of the difference problem on the grid of any level except the roughest one. The operator S p turns the current error e 0 = v − u for p smoothing steps to the error e 1 = v − u, i.e., e 1 = S p e 0 .…”

“…The amplitude of such components decreases almost inversely proportional to the frequency, which corresponds to the needs of the multigrid method. As a result, for the geometric classical multigrid method (MM) on Cartesian grids, we have constructed an effective parallel adaptive implementation (see [18][19][20][21]); now we develop it for the algebraic multigrid method.…”

Многосеточные Методы С Косо-Эрмитовыми Сглаживателями Для Задач Конвекции–диффузии С Преобладающей Конвекцией

“…The GMG for the elliptic problems with anisotropic discontinuous coefficients has been investigated in [15]. The authors of this paper have considered a 3-D diffusion problem with general boundary conditions and studied two iterative smoothers: the Chebyshev operator polynomial and a rational function.…”