Domain splitting algorithm for mixed finite element approximations to parabolic problems.

*(English)*Zbl 0878.65081A domain decomposition algorithm for solving the mixed finite element approximations to parabolic initial-boundary value problems is proposed. In contrast to the ordinary overlapping domain decomposition method this technique leads to noniterative algorithms, i.e. the subdomain problems are solved independently and the solution in the whole domain is obtained from the local solutions by restriction and simple averaging.

The time discretization leads to an elliptic problem with large positive coefficient of the zero order term. The solutions of such problems exhibit a boundary layer with thickness proportional to the square root of the time discretization parameter. Thus, any error in the boundary conditions decays exponentially and a reasonable overlap produces a sufficiently accurate method. It is proved that the proposed algorithm is stable in \(L\)-norm and has the same accuracy as the implicit method.

The time discretization leads to an elliptic problem with large positive coefficient of the zero order term. The solutions of such problems exhibit a boundary layer with thickness proportional to the square root of the time discretization parameter. Thus, any error in the boundary conditions decays exponentially and a reasonable overlap produces a sufficiently accurate method. It is proved that the proposed algorithm is stable in \(L\)-norm and has the same accuracy as the implicit method.

Reviewer: R.Anguelov (Silverton)

##### MSC:

65M55 | Multigrid methods; domain decomposition 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 |