# Solving the Incompressible Surface Stokes Equation by Standard Velocity-Correction Projection Methods

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Surface Differential Operator

**Remark**

**1.**

## 3. Finite Element Approximation of Surface

## 4. Standard Velocity Correction Method

#### 4.1. First-Order Velocity Correction Method

**Step 1.**We solve for ${\mathbf{u}}^{k+1}\in {V}_{T},{p}^{k+1}\in Q$ from

**Step 2**. We solve for ${\tilde{\mathbf{u}}}^{k+1}\in {V}_{T}$ from

**Implementation of the standard form**

**Step 1**. We solve for ${\mathbf{u}}^{k+1}\in {V}_{T},{p}^{k+1}\in Q$ from

**Step 2**. We solve for ${\tilde{\mathbf{u}}}^{k+1}\in {V}_{T}$ from

#### 4.2. Second-Order Velocity Correction Method

**Step 1**. We solve for ${\mathbf{u}}^{k+1}\in {V}_{T},{p}^{k+1}\in Q$ from

**Step 2**. We solve for ${\tilde{\mathbf{u}}}^{k+1}\in {V}_{T}$ from

**Implementation of the standard form**

**Step 1**. We solve for ${\mathbf{u}}^{k+1}\in {V}_{T},{p}^{k+1}\in Q$ from

**Step 2**. We solve ${\tilde{\mathbf{u}}}^{k+1}\in {V}_{T}$ from

#### 4.3. Stability Analysis

#### 4.3.1. First-Order Scheme Stability Analysis

**Theorem**

**1.**

**Proof.**

#### 4.3.2. Stability Analysis of Second-Order Schemes

**Theorem**

**2.**

**Proof.**

#### 4.4. Fully Discrete Scheme

#### 4.4.1. First-Order Fully Discrete Velocity Correction Method

**Step 1**. For $\forall {v}_{ih}\in {V}_{Th},\forall {q}_{h}\in {Q}_{h}$, we solve for $({\mathbf{u}}_{h}^{k+1},{p}_{h}^{k+1})$ from

**Step 2**. For $\forall {v}_{ih}\in {V}_{T}$, we solve for ${\tilde{\mathbf{u}}}_{h}^{k+1}\in {V}_{Th}$ from

#### 4.4.2. Second-Order Fully Discrete Velocity Correction Method

**Step 1**. For $\forall {v}_{ih}\in {V}_{Th},\forall {q}_{h}\in {Q}_{h}$, we solve for $({\mathbf{u}}_{h}^{k+1},{p}_{h}^{k+1})$ from

**Step 2**. For $\forall {v}_{ih}\in {V}_{T}$, we solve for ${\tilde{\mathbf{u}}}_{h}^{k+1}\in {V}_{Th}$ from

## 5. Numerical Experiments

#### 5.1. Convergence Test

#### 5.2. A Circular Flow on a Ring

#### 5.3. Stability Test

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Zhao, Y.; Feng, X.
Solving the Incompressible Surface Stokes Equation by Standard Velocity-Correction Projection Methods. *Entropy* **2022**, *24*, 1338.
https://doi.org/10.3390/e24101338

**AMA Style**

Zhao Y, Feng X.
Solving the Incompressible Surface Stokes Equation by Standard Velocity-Correction Projection Methods. *Entropy*. 2022; 24(10):1338.
https://doi.org/10.3390/e24101338

**Chicago/Turabian Style**

Zhao, Yanzi, and Xinlong Feng.
2022. "Solving the Incompressible Surface Stokes Equation by Standard Velocity-Correction Projection Methods" *Entropy* 24, no. 10: 1338.
https://doi.org/10.3390/e24101338