A posteriori error estimates for the generalized overlapping domain decomposition method for a parabolic variational equation with mixed boundary condition

Salah Boulaaras,Khaled Habita,Mohamed Haiour

doi:10.5269/bspm.v38i4.40535

Abstract

The paper deals with a posteriori error estimates for the generalized overlapping domain decomposition method with mixed boundary condition the interfaces for parabolic variational equation with Laplace boundary value problems are proved using by the theta time scheme combined with Galerkin spatial approximation.

Highlights

We develop an approach which combines a result of geometrical convergence due to [6], [17], [18] and a lemma which consists of estimating the error in the uniform norm between the continuous and discrete Schwarz iterates
In [21] the authors presented the error analysis in the maximum norm for a class of nonlinear elliptic problems in the context of overlapping nonmatching grids and they studied the optimal error estimate on uniform norm between the discrete Schwarz sequence and the exact solution of the partial differential equations, and in [22] the authors derived a posteriori error estimates for generalized overlapping domain decomposition method (GODDM) with Direchlet boundary conditions on the interfaces for Laplace boundary value problems, they have proved that the error estimate in the continuous case depends on the differences of the traces of the subdomain solutions on the interfaces using Galerkin method
In [7], we have treated the overlapping domain decomposition method combined with a finite element approximation for elliptic equation related for Laplace operator ∆, where a Sobolev norm analysis of an overlapping Schwarz method on nonmatching grids has been used, where we proved that the discretization on every subdomain converges in Soblev norm

Summary

The continuous problem

The problem (1.1) can be reformulated into the following continuous parabolic variational equation: find u ∈ L2 0, T, H01 (Ω) solution of. We extend the last work for parabolic equation with mixed boundary conditions where we prove an a posteriori error estimates for the generalized overlapping domain decomposition method with mixed boundary conditions on the boundaries for the discrete solutions on subdomains using theta time scheme combined with a finite element spatial approximation, similar to that in [21], which investigated full elliptic operator with Dirichlet boundary condition.

The space discretization

The space continuous for the generalized Schwarz method

A posteriori error estimate in continuous case

A Posteriori Error Estimate

Conclusion