Domain decomposition methods for unsteady convection-diffusion problems.

*(English)*Zbl 0744.65063
Computing methods in applied sciences and engineering, Proc. 9th Int. Conf., Paris/Fr. 1990, 211-227 (1990).

[For the entire collection see Zbl 0724.00024.]

The article is an observation of earlier results obtained by the author for the domain decomposition (DD) method with alternating Neumann- Dirichlet boundary conditions applied to the specific system \(Aw=g\) of linear finite element algebraic equations. This system arises at each time step while the second order parabolic partial differential equation is approximated by an implicit finite difference scheme in time and by means of the finite element method in the space variables. As a model problem the convection-diffusion equation in an arbitrary sufficiently smooth domain is considered.

First of all conditions are formulated which guarantee that the two-level DD- preconditioner is equivalent in spectrum to the matrix \(A\). The author also proves that the mesh Green function for the mesh operator \(A\) decreases as \(\exp(-c| x-y| /\sqrt{\Delta t})\), \(c=\hbox{const}>0\), when the point \(x\) is moving away from the point \(y\) of the location of the source. This allows him to suggest a non-iterative DD-method with overlapping sufficiently small subdomains.

The article is an observation of earlier results obtained by the author for the domain decomposition (DD) method with alternating Neumann- Dirichlet boundary conditions applied to the specific system \(Aw=g\) of linear finite element algebraic equations. This system arises at each time step while the second order parabolic partial differential equation is approximated by an implicit finite difference scheme in time and by means of the finite element method in the space variables. As a model problem the convection-diffusion equation in an arbitrary sufficiently smooth domain is considered.

First of all conditions are formulated which guarantee that the two-level DD- preconditioner is equivalent in spectrum to the matrix \(A\). The author also proves that the mesh Green function for the mesh operator \(A\) decreases as \(\exp(-c| x-y| /\sqrt{\Delta t})\), \(c=\hbox{const}>0\), when the point \(x\) is moving away from the point \(y\) of the location of the source. This allows him to suggest a non-iterative DD-method with overlapping sufficiently small subdomains.

Reviewer: V.Korneev (St.Petersburg)

##### MSC:

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |