# On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring

## Abstract

**:**

## 1. Introduction

## 2. General Convergence Result

**Assumption**

**1.**

**Assumption**

**2.**

**Algorithm**

**1.**

**Algorithm**

**2.**

**Remark**

**1.**

**Theorem**

**1.**

- (i)
- if $p=q=2$ we haven

- (ii)
- if $p>q$ we have

**Proof.**

**Remark**

**2.**

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

## 3. Damped Additive Schwarz Methods in Finite Element Spaces

**Remark**

**4.**

## 4. Numerical Results

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Number of iterations depending on the overlap parameter: damping parameter associated with the color (

**left**) and constant damping parameter (

**right**).

**Figure 4.**Sections in the direction of a coordinate axis passing through the center of the functions of a unity partition: $\delta =0.01$ (

**left**) and $\delta =0.14$ (

**right**).

**Figure 5.**Sections in the direction of a coordinate axis passing through the center of the functions of a unity partition: $\delta =0.15$ (

**left**) and $\delta =0.18$ (

**right**).

**Figure 6.**Number of iterations depending on the number of subdomains: damping parameter associated with the color (

**left**) and constant damping parameter (

**right**).

**Figure 7.**Sections in the direction of a coordinate axis passing through the center of the functions of a unity partition: ${2}^{2}$ subdomains (

**left**) and ${24}^{2}$ subdomains (

**right**).

**Figure 8.**Sections in the direction of a coordinate axis passing through the center of the functions of a unity partition: ${25}^{2}$ subdomains (

**left**) and ${48}^{2}$ subdomains (

**right**).

**Table 1.**Number of iterations depending on the various damping parameters associated with the colors.

${\mathit{\varrho}}_{1}$ | ${\mathit{\varrho}}_{2}$ | ${\mathit{\varrho}}_{3}$ | ${\mathit{\varrho}}_{4}$ | Number of Iterations |
---|---|---|---|---|

0.10 | 0.10 | 0.35 | 0.45 | 451 |

0.10 | 0.15 | 0.25 | 0.50 | 406 |

0.10 | 0.15 | 0.40 | 0.35 | 396 |

0.10 | 0.20 | 0.30 | 0.40 | 377 |

0.10 | 0.25 | 0.20 | 0.45 | 383 |

0.10 | 0.25 | 0.35 | 0.30 | 365 |

0.10 | 0.30 | 0.25 | 0.35 | 366 |

0.20 | 0.20 | 0.55 | 0.35 | 292 |

0.20 | 0.30 | 0.20 | 0.30 | 288 |

0.25 | 0.25 | 0.25 | 0.25 | 275 |

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**MDPI and ACS Style**

Badea, L.
On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring. *Math. Comput. Appl.* **2022**, *27*, 59.
https://doi.org/10.3390/mca27040059

**AMA Style**

Badea L.
On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring. *Mathematical and Computational Applications*. 2022; 27(4):59.
https://doi.org/10.3390/mca27040059

**Chicago/Turabian Style**

Badea, Lori.
2022. "On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring" *Mathematical and Computational Applications* 27, no. 4: 59.
https://doi.org/10.3390/mca27040059