Parallel algorithm for convection-diffusion system based on least-squares procedure.

Jiansong Zhang,Hui Guo,Hongfei Fu,Yanzhen Chang

doi:10.1186/s40064-016-3333-8

Abstract

Combining subspace correction method with least-squares finite element procedure, we construct a new overlapping domain decomposition parallel algorithm for solving the first-order time-dependent convection–diffusion system. This algorithm is fully parallel. We analyze the convergence of approximate solution, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration number and sub-domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to given accuracy at each time step.

Highlights

In this paper, we consider the following initial-boundary value problem for timedependent convection–diffusion system: c ∂u ∂t + ∇ · σ qu = f, x ∈ Ω, 0 < t ≤ T, A∇u bu0, x ∈ Ω, 0 < t ≤ T, u = 0, x ∈ ΓD, (1)
We analyze the convergence of approximate solution, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration number and sub-domains overlapping degree
Both theoretical analysis and numerical experiments indicate the full parallelization of the algorithms and very good approximate property

Summary

Background

We consider the following initial-boundary value problem for timedependent convection–diffusion system: c. Proof of Theorem 1 It is seen that parallel algorithm is equivalent to use an iteration with initial values (σhn−1, unh−1) to solve the following equation: (σhn, unh) ∈ Whσ × Vhu such that for any (ωh, vh) ∈ Whσ × Vhu an((σhn, unh), (ωh, vh)) =. Lemma 6 For parallel algorithm, we have the following estimate (σhn − σhn−1, unh − unh−1) 2an tn. These numerical results suggest that we can get a good result for convection–diffusion problem using parallel algorithm , even iterating only one or two cycle at each time step These numerical results imply that the errors caused by decomposing domain decrease as the discretization parameters h and τ decrease and increase as the overlapping degree H becomes small, which are coincided with our theoretical result.

Conclusions

Methods