# Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

**Remark**

**1.**

## 3. Fine Grid Approximation

## 4. Coarse Grid Approximation

**Remark**

**2.**

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the heterogeneous property, coarse grid, and multiscale basis functions in the local domain.

**Figure 4.**Fine grid solution (

**top**) and multiscale solution using eight multiscale basis functions (

**bottom**). Coefficient k from Test 1 with $\beta =0$.

**Figure 5.**Fine grid solution (

**top**) and multiscale solution using eight multiscale basis functions (

**bottom**). Coefficient k from Test 2 with $C=34.93$.

**Figure 6.**Fine grid solution (

**top**) and multiscale solution using eight multiscale basis functions (

**bottom**). Coefficient k from Test 1 with $C=34.93$.

**Figure 7.**Fine grid solution (

**top**) and multiscale solution using eight multiscale basis functions (

**bottom**). Coefficient k from Test 2 with $\beta =0$.

**Table 1.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with $\beta =0$.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % |
---|---|---|---|---|---|---|---|

1 | 320 | 10.069 | 1.212 | 1 | 320 | 11.279 | 1.451 |

2 | 540 | 1.112 | 0.031 | 2 | 540 | 2.943 | 0.104 |

4 | 980 | 0.253 | 0.001 | 4 | 980 | 0.579 | 0.004 |

8 | 1860 | 0.061 | 0.001 | 8 | 1860 | 0.152 | 0.001 |

**Table 2.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with $C=10.24$.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter |
---|---|---|---|---|---|---|---|---|---|

1 | 320 | 10.334 | 1.856 | 9 | 1 | 320 | 11.042 | 2.155 | 7 |

2 | 540 | 2.329 | 0.136 | 8 | 2 | 540 | 3.585 | 0.252 | 6 |

4 | 980 | 2.174 | 0.129 | 7 | 4 | 980 | 2.897 | 0.199 | 5 |

8 | 1860 | 2.054 | 0.102 | 6 | 8 | 1860 | 2.856 | 0.195 | 5 |

Iterations on the fine grid = 5 | Iterations on the fine grid = 5 |

**Table 3.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with $C=34.93$.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter |
---|---|---|---|---|---|---|---|---|---|

1 | 320 | 10.389 | 2.052 | 24 | 1 | 320 | 11.021 | 2.374 | 22 |

2 | 540 | 2.766 | 0.171 | 23 | 2 | 540 | 4.059 | 0.319 | 20 |

4 | 980 | 2.567 | 0.151 | 21 | 4 | 980 | 3.469 | 0.248 | 19 |

8 | 1860 | 2.561 | 0.151 | 19 | 8 | 1860 | 3.446 | 0.244 | 17 |

Iterations on the fine grid = 21 | Iterations on the fine grid = 21 |

**Table 4.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with $C=1581.14$.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter |
---|---|---|---|---|---|---|---|---|---|

1 | 320 | 10.415 | 2.181 | 387 | 1 | 320 | 11.033 | 2.518 | 399 |

2 | 540 | 2.998 | 0.197 | 403 | 2 | 540 | 4.364 | 0.364 | 388 |

4 | 980 | 2.814 | 0.177 | 389 | 4 | 980 | 3.868 | 0.294 | 371 |

8 | 1860 | 2.799 | 0.175 | 368 | 8 | 1860 | 3.843 | 0.291 | 348 |

Iterations on the fine grid = 1162 | Iterations on the fine grid = 1143 |

**Table 5.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with C = 71,554.17.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter |
---|---|---|---|---|---|---|---|---|---|

1 | 320 | 10.416 | 2.185 | 744 | 1 | 320 | 11.033 | 2.522 | 794 |

2 | 540 | 3.003 | 0.198 | 854 | 2 | 540 | 4.372 | 0.365 | 809 |

4 | 980 | 2.821 | 0.177 | 886 | 4 | 980 | 3.878 | 0.295 | 805 |

8 | 1860 | 2.805 | 0.176 | 864 | 8 | 1860 | 3.853 | 0.291 | 777 |

Iterations on the fine grid = 16,489 | Iterations on the fine grid = 14,466 |

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## Share and Cite

**MDPI and ACS Style**

Spiridonov, D.; Huang, J.; Vasilyeva, M.; Huang, Y.; Chung, E.T.
Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model. *Mathematics* **2019**, *7*, 1212.
https://doi.org/10.3390/math7121212

**AMA Style**

Spiridonov D, Huang J, Vasilyeva M, Huang Y, Chung ET.
Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model. *Mathematics*. 2019; 7(12):1212.
https://doi.org/10.3390/math7121212

**Chicago/Turabian Style**

Spiridonov, Denis, Jian Huang, Maria Vasilyeva, Yunqing Huang, and Eric T. Chung.
2019. "Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model" *Mathematics* 7, no. 12: 1212.
https://doi.org/10.3390/math7121212