Finite Element Method for a Fractional-Order Filtration Equation with a Transient Filtration Law
Abstract
:1. Introduction
2. Materials and Methods
2.1. Formulation of the Problem
2.2. The Uniqueness of the Solution and Its Continuous Dependence on the Input Data
2.2.1. Weak Variational Formulation of the Problem
2.2.2. Uniqueness of the Solution and Its Continuous Dependence on the Input Data
2.3. A Numerical Method
2.3.1. Construction of a Semi-Discrete Numerical Scheme
2.3.2. Construction of the Fully Discrete Scheme
2.4. Stability of the Numerical Scheme
2.5. Convergence of the Numerical Method
2.5.1. Convergence of a Semi-Discrete Scheme
2.5.2. Convergence of a Fully Discrete Scheme
2.5.3. Convergence of the Iterative Process
3. Results
Verification of the Order of Convergence Based on Computational Experiments
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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-Error | Order | -Error | Order | -Error | Order | |
---|---|---|---|---|---|---|
1/10 | - | - | - | |||
1/20 | 3.75 | 3.69 | 3.63 | |||
1/30 | 3.78 | 3.72 | 3.65 | |||
1/40 | 3.80 | 3.73 | 3.66 | |||
1/50 | 3.81 | 3.74 | 3.66 | |||
1/60 | 3.83 | 3.75 | 3.67 | |||
1/70 | 3.85 | 3.76 | 3.67 | |||
1/80 | 3.86 | 3.77 | 3.68 | |||
Predicted | 3.9 | 3.8 | 3.7 | |||
-Error | Order | -Error | Order | -Error | Order | |
1/10 | - | - | - | |||
1/20 | 3.55 | 3.47 | 3.38 | |||
1/30 | 3.57 | 3.48 | 3.39 | |||
1/40 | 3.57 | 3.48 | 3.39 | |||
1/50 | 3.58 | 3.49 | 3.39 | |||
1/60 | 3.58 | 3.49 | 3.39 | |||
1/70 | 3.58 | 3.49 | 3.39 | |||
1/80 | 3.58 | 3.49 | 3.40 | |||
Predicted | 3.6 | 3.5 | 3.4 | |||
-Error | Order | -Error | Order | -Error | Order | |
1/10 | - | - | - | |||
1/20 | 3.29 | 3.19 | 3.11 | |||
1/30 | 3.29 | 3.20 | 3.10 | |||
1/40 | 3.29 | 3.20 | 3.10 | |||
1/50 | 3.30 | 3.20 | 3.10 | |||
1/60 | 3.30 | 3.20 | 3.10 | |||
1/70 | 3.30 | 3.20 | 3.10 | |||
1/80 | 3.30 | 3.20 | 3.10 | |||
Predicted | 3.3 | 3.2 | 3.1 |
Mesh Parameters | |||||
---|---|---|---|---|---|
Nodes | 441 | 1681 | 10,201 | 40,401 | 160,801 |
Cells | 100 | 400 | 2500 | 10,000 | 40,000 |
CPU Time (s) | |||||
1/10 | 0.0322 | 0.1113 | 1.0653 | 3.6686 | 16.0136 |
1/20 | 0.0828 | 0.2362 | 1.2875 | 6.2702 | 30.8062 |
1/30 | 0.1233 | 0.3537 | 1.9482 | 9.3582 | 46.3597 |
1/40 | 0.1564 | 0.4630 | 2.5717 | 12.3262 | 61.7358 |
1/50 | 0.1866 | 0.5783 | 3.1718 | 15.0545 | 76.6795 |
1/60 | 0.2306 | 0.6953 | 3.8219 | 18.0586 | 91.8569 |
1/70 | 0.2624 | 0.8053 | 4.5044 | 21.4017 | 106.8736 |
1/80 | 0.2985 | 0.9102 | 5.1263 | 24.2711 | 122.1779 |
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Alimbekova, N.; Berdyshev, A.; Madiyarov, M.; Yergaliyev, Y. Finite Element Method for a Fractional-Order Filtration Equation with a Transient Filtration Law. Mathematics 2024, 12, 2519. https://doi.org/10.3390/math12162519
Alimbekova N, Berdyshev A, Madiyarov M, Yergaliyev Y. Finite Element Method for a Fractional-Order Filtration Equation with a Transient Filtration Law. Mathematics. 2024; 12(16):2519. https://doi.org/10.3390/math12162519
Chicago/Turabian StyleAlimbekova, Nurlana, Abdumauvlen Berdyshev, Muratkan Madiyarov, and Yerlan Yergaliyev. 2024. "Finite Element Method for a Fractional-Order Filtration Equation with a Transient Filtration Law" Mathematics 12, no. 16: 2519. https://doi.org/10.3390/math12162519