Hermite Finite Element Method for One-Dimensional Fourth-Order Boundary Value Problems
Abstract
:1. Introduction
2. One-Dimensional Fourth-Order Boundary Value Problems
2.1. Preliminaries
2.2. Problem Model
2.3. Weak Formulation
3. Finite Element Approximation
3.1. Hermite Finite Element Method
- Hermite cubic basis function:
- Hermite quintic basis function:
3.2. Discrete Scheme
4. Error Estimations
5. Numerical Simulations
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Mesh Size (h) | Semi-Error | Error Order | Error | Error Order |
---|---|---|---|---|
1 | 1.1405 | 0.0499 | ||
1/2 | 0.2843 | 2.0043 | 0.0031 | 4.0035 |
1/4 | 0.0710 | 2.0017 | 1.9405 × 10 | 4.0042 |
1/8 | 0.0177 | 2.0004 | 1.2118 × 10 | 4.0013 |
1/16 | 0.0044 | 2.0001 | 7.5718 × 10 | 4.0003 |
1/32 | 0.0010 | 2.0000 | 4.7220 × 10 | 4.0031 |
Mesh Size (h) | Semi-Error | Error Order | Error | Error Order |
---|---|---|---|---|
1 | 1.1421 | 0.0378 | ||
1/2 | 0.2843 | 2.0062 | 0.0024 | 3.9537 |
1/4 | 0.0710 | 2.0019 | 1.5303 × 10 | 3.9945 |
1/8 | 0.0177 | 2.0004 | 9.5705 × 10 | 3.9990 |
1/16 | 0.0044 | 2.0001 | 5.9824 × 10 | 3.9998 |
1/32 | 0.0011 | 2.0000 | 3.7066 × 10 | 4.0125 |
Mesh Size (h) | Semi-Error | Error Order | Error | Error Order |
---|---|---|---|---|
1 | 0.0081 | 7.7134 × 10 | ||
1/2 | 5.7698 × 10 | 3.8129 | 1.3077 × 10 | 5.8852 |
1/4 | 3.6226 × 10 | 3.9934 | 1.9897 × 10 | 6.0384 |
1/8 | 2.2574 × 10 | 4.0034 | 3.0660 × 10 | 6.0020 |
1/16 | 1.4090 × 10 | 4.0019 | 4.5808 × 10 | 2.7427 |
Mesh Size (h) | Semi-Error | Error Order | Error | Error Order |
---|---|---|---|---|
1 | 0.0081 | 7.7198 × 10 | ||
1/2 | 5.7698 × 10 | 3.8130 | 1.3078 × 10 | 5.8833 |
1/4 | 3.6226 × 10 | 3.9934 | 1.9897 × 10 | 6.0385 |
1/8 | 2.2574 × 10 | 4.0043 | 3.0662 × 10 | 6.0200 |
1/16 | 1.4090 × 10 | 4.0019 | 1.5879 × 10 | 0.9493 |
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Wu, B.; Qiu, J. Hermite Finite Element Method for One-Dimensional Fourth-Order Boundary Value Problems. Mathematics 2024, 12, 1613. https://doi.org/10.3390/math12111613
Wu B, Qiu J. Hermite Finite Element Method for One-Dimensional Fourth-Order Boundary Value Problems. Mathematics. 2024; 12(11):1613. https://doi.org/10.3390/math12111613
Chicago/Turabian StyleWu, Bangmin, and Jiali Qiu. 2024. "Hermite Finite Element Method for One-Dimensional Fourth-Order Boundary Value Problems" Mathematics 12, no. 11: 1613. https://doi.org/10.3390/math12111613
APA StyleWu, B., & Qiu, J. (2024). Hermite Finite Element Method for One-Dimensional Fourth-Order Boundary Value Problems. Mathematics, 12(11), 1613. https://doi.org/10.3390/math12111613