# Rational Approximations in Robust Preconditioning of Multiphysics Problems

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Monolithically Coupled Multiphysics Problems: Two Examples

#### 2.1. Example 1: Coupling of Darcy and Stokes Flows

#### 2.2. Example 2: Coupling a 1D Structure Embedded in $\mathrm{\Omega}\subset {\mathbb{R}}^{3}$

#### 2.3. Implementation of ${C}_{D-S}$ and ${C}_{3-1}$: The Key Question

## 3. The BURA Method

## 4. Bura Preconditioning: Condition Number Estimates

#### 4.1. Bura Preconditioner ${C}_{\alpha ,k}^{BURA}$: $\alpha \in (0,1)$

**Lemma**

**1.**

**Proof.**

#### 4.2. Bura Preconditioner ${C}_{\beta ,k}^{BURA}$: $\beta \in (-1,0)$

**Lemma**

**2.**

**Proof.**

## 5. Preconditioning of the Coupled Problems: Condition Number Estimates and Computational Complexity

- (A1)
- Quasi uniform meshes with mesh parameter h are used for FDM/FEM approximation of the considered coupled problems. Thus, the number of unknowns for the discrete Darcy–Stokes problem is ${N}_{D-S}=O\left({h}^{-2}\right)$, and similarly, for the 1D–3D coupling problem, ${N}_{3-1}=O\left({h}^{-3}\right)$. In both cases, the number of unknowns related to the interface $\mathrm{\Gamma}$ is ${N}_{\mathrm{\Gamma}}=O\left({h}^{-1}\right)$.
- (A2)
- Solvers of optimal computational complexity are applied for the systems with sparse SPD matrices which appear in the implementation of the BURA preconditioners.

#### 5.1. Preconditioner ${\mathbb{C}}_{D-S}^{BURA}$

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 5.2. Preconditioner ${\mathbb{C}}_{3-1}^{BURA}$

**Lemma**

**4.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 6. Behavior of the Condition Numbers of the BURA-Based Preconditioners

#### 6.1. Analysis of $\kappa \left({\left[{\mathbb{C}}_{0.5,k}^{BURA}\right]}^{-1}{\mathbb{A}}^{0.5}\right)$

#### 6.2. Analysis of $\kappa \left({\left[{\mathbb{C}}_{-0.14,k}^{BURA}\right]}^{-1}{\mathbb{A}}^{-0.14}\right)$

#### 6.3. Comparative Summary

## 7. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Darcy–Stokes system in a rectangle domain $\mathrm{\Omega}={\mathrm{\Omega}}_{S}\cup {\mathrm{\Omega}}_{D}$: model problem with simple geometry (

**left**) and more realistic problem with a more general geometry of the interface (

**right**).

**Figure 2.**Coupled 3D–1D systems in a hexahedral domain $\mathrm{\Omega}$: model problems with single line geometry of the interfaces (

**left**) and more realistic problem with more general geometry of the interface (

**right**).

**Table 1.**Computed estimates of $\kappa \left({\left[{\mathbb{C}}_{0.5,k}^{BURA}\right]}^{-1}{\mathbb{A}}^{0.5}\right)$ and $\kappa \left({\left[{\mathbb{C}}_{-0.14,k}^{BURA}\right]}^{-1}{\mathbb{A}}^{-0.14}\right)$ with respect to $3\le k\le 9$, for $\delta =\kappa \left(\mathbb{A}\right)\in \{{10}^{5},{10}^{6},{10}^{7},{10}^{8}\}$.

k | $\mathit{\alpha}=0.5$ | $\mathit{\beta}=-0.14$ | ||||||
---|---|---|---|---|---|---|---|---|

$\mathit{\delta}={\mathbf{10}}^{\mathbf{5}}$ | $\mathit{\delta}={\mathbf{10}}^{\mathbf{6}}$ | $\mathit{\delta}={\mathbf{10}}^{\mathbf{7}}$ | $\mathit{\delta}={\mathbf{10}}^{\mathbf{8}}$ | $\mathit{\delta}={\mathbf{10}}^{\mathbf{5}}$ | $\mathit{\delta}={\mathbf{10}}^{\mathbf{6}}$ | $\mathit{\delta}={\mathbf{10}}^{\mathbf{7}}$ | $\mathit{\delta}={\mathbf{10}}^{\mathbf{8}}$ | |

3 | 1.46 | 3.23 | 9.98 | 31.50 | 3.74 | 23.69 | 168.62 | 1221.51 |

4 | 1.38 | 1.46 | 3.29 | 10.19 | 1.38 | 5.64 | 37.57 | 270.26 |

5 | 1.08 | 1.43 | 1.46 | 3.78 | 1.09 | 1.96 | 10.20 | 71.12 |

6 | 1.03 | 1.08 | 1.46 | 1.66 | 1.06 | 1.13 | 3.43 | 21.38 |

7 | 1.02 | 1.06 | 1.18 | 1.46 | 1.01 | 1.09 | 1.58 | 7.31 |

8 | 1.01 | 1.03 | 1.08 | 1.34 | 1.01 | 1.05 | 1.09 | 2.95 |

9 | 1.00 | 1.01 | 1.03 | 1.08 | 1.00 | 1.01 | 1.09 | 1.54 |

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**MDPI and ACS Style**

Harizanov, S.; Lirkov, I.; Margenov, S.
Rational Approximations in Robust Preconditioning of Multiphysics Problems. *Mathematics* **2022**, *10*, 780.
https://doi.org/10.3390/math10050780

**AMA Style**

Harizanov S, Lirkov I, Margenov S.
Rational Approximations in Robust Preconditioning of Multiphysics Problems. *Mathematics*. 2022; 10(5):780.
https://doi.org/10.3390/math10050780

**Chicago/Turabian Style**

Harizanov, Stanislav, Ivan Lirkov, and Svetozar Margenov.
2022. "Rational Approximations in Robust Preconditioning of Multiphysics Problems" *Mathematics* 10, no. 5: 780.
https://doi.org/10.3390/math10050780