Keywords: multigrid method, finite difference method, V-cycle, linear systems, approximate inverse smoothing, dynamic over-under relaxation method, preconditioned conjugate gradient method, initial value problems.

Many engineering and scientific problems are described using sparse unsymmetric linear systems derived from the discretization of elliptic and parabolic equations in two space variables. This category of equations represents a large class of commonly occurring problems in mathematical physics and engineering (i.e. see-page flow or irrotational hydrodynamics flow problems, diffusion theory and plasma physics problems, etc.).

The purpose of this work is the derivation of hybrid heterogeneous schemes based on time implicit approximating schemes, namely backward differences and the Crank-Nickolson method, with the multigrid method in conjunction with explicit approximate inverses for the efficient iterative solution of unsymmetric linear systems.

Multigrid techniques have gained the interest of the research community during the recent decades. Multigrid methods are based in the observation that the low frequency components of the error are increasing as the resolution of the discretization deteriorates. The high frequency components of the error are effectively minimized within the first few iterations of the stationary iterative solvers, while the slow convergence of the stationary iterative solvers is the result of the presence of persistent low frequency error modes.

The major components of the multigrid method are: smoother, prolongation operator (P), restriction operator (R), cycle strategy. Smoothers are often stationary iterative methods (for example: the Jacobi method, first order Richardson method). Specifically, the first order explicit preconditioned Richardson method has been used as a smoother in conjunction with explicit approximate inverses. The prolongation operator is in fact an interpolation procedure that transfers error approximation to finer grids. The prolongation operator used was the two-dimensional linear interpolant. Restriction operators are used to transfer vectors from finer to coarser grids. The cycle strategy is refers to the descending and ascending phase resulting in V, W or F cycles. In our approach the V cycle strategy is selected.

An important achievement over recent decades is the appearance and use of approximate inverse preconditioning methods for solving elliptic and parabolic partial differential equations. The effectiveness of the preconditioning methods relies on the construction and use of an efficient preconditioner, factors in the sense that the preconditioners are close approximants to the coefficient matrix.

In addition, a new class of Krylov subspace iterative methods in conjunction with multigrid preconditioning based on explicit approximate inverses is used for solving parabolic partial differential equations.

Finally, the applicability of the new proposed hybrid heterogeneous implicit time-explicit approximate inverses combined with multigrid methods for parabolic PDEs is discussed by solving characteristic time-dependent problems in two dimensions and the numerical results are given in the paper

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