Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary
Abstract
:1. Introduction
2. Problem Formulation
- (a)
- (b)
- () are differentiable functions on , such that the inequalities
- (c)
- the function satisfies the inequalities
3. The Scheme of the Finite Element Method
4. Numerical Experiments
5. Conclusions
- to reduce the absolute value of the error it is more expedient to increase the number of layers n than to reduce the width of the boundary ring domain; in this case the absolute value of the error decreases faster;
- for meshes of large dimensionality it is advisable to use the weighted finite element method.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Quasi-Uniform Mesh () | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |
Number of nodes N | 10,849,474 | | | 0.00% | 69 |
| 89.37% | 9,696,362 | |||
| 10.61% | 1,151,232 | |||
h | 0.00055 | | 0.01% | 1196 | |
| 0.00% | 372 | |||
| 0.00% | 243 | |||
Refined Mesh () | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |
N | 10,755,478 | | | 0.00% | 0 |
Number of nodes in domain | 10,661,162 | | 0.00% | 0 | |
| 0.00% | 4 | |||
h | 0.000556 | | 44.46% | 4,781,879 | |
n | 3 | | 55.44% | 5,962,761 | |
b | 1/1024 | | 0.10% | 10,834 | |
Refined Mesh () | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |
N | 4,974,486 | | | 0.00% | 0 |
Number of nodes in domain | 4,241,164 | | 0.00% | 0 | |
| 0.00% | 0 | |||
h | 0.00087 | | 2.75% | 136,821 | |
n | 18 | | 83.18% | 4,137,684 | |
b | 1/128 | | 14.07% | 699,981 |
Quasi-Uniform Mesh () | Refined Mesh (), | ||||||||
---|---|---|---|---|---|---|---|---|---|
0.0022 | 0.0035 | ||||||||
1.68 | 1.85 | 4.21 | 2.06 | ||||||
0.0011 | 0.00169 | ||||||||
1.83 | 1.72 | 4.11 | 2.03 | ||||||
0.00055 | 0.00083 |
Specified Limited Error | Quasi-Uniform Mesh () | ||
| | | |
| |||
| |||
| |||
| |||
| |||
N | 338,449 | 1,355,498 | 5,427,739 |
h | 0.0031 | 0.0015 | 0.00078 |
Specified Limited Error | Refined Mesh (), | ||
| | | |
| |||
| |||
| |||
| |||
| |||
N | 425,760 | 1,569,052 | 6,129,755 |
Number of nodes in domain | 386,628 | 1,375,684 | 5,200,079 |
h in domain | 0.0029 | 0.0015 | 0.00079 |
Quasi-Uniform Mesh () | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |
N | 2,713,152 | | | 50.40% | 1,367,499 |
| 49.60% | 1,345,653 | |||
| 0.00% | 0 | |||
h | 0.00055 | | 0.00% | 0 | |
| 0.00% | 0 | |||
Refined Mesh () | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |
N | 3,254,432 | | | 0.00% | 0 |
Number of nodes in domain | 2,484,744 | | 0.00% | 0 | |
h in domain | 0.0011 | | 26.32% | 856,521 | |
n | 27 | | 31.94% | 1,039,571 | |
b | | 41.74% | 1,358,340 |
Quasi-Uniform Mesh () | Refined Mesh (), | |||||
---|---|---|---|---|---|---|
in domain | ||||||
0.0022 | 0.065816 | 3 | 0.0068 | 0.04562 | ||
1.43 | 2.08 | |||||
0.0011 | 0.046016 | 8 | 0.0032 | 0.021939 | ||
1.43 | 2.00 | |||||
0.00055 | 0.032220 | 18 | 0.0016 | 0.010989 |
Specified Limited Error | Quasi-Uniform Mesh () | ||
| | | |
| |||
| |||
| |||
| |||
N | 168,670 | 677,704 | 2,713,152 |
h | 0.0044 | 0.0022 | 0.0011 |
Specified Limited Error | Refined Mesh () | ||
| | | |
| |||
| |||
| |||
| |||
N | 166,751 | 643,498 | 2,567,442 |
Number of nodes in domain | 139,635 | 514,952 | 1,972,944 |
h in domain | 0.0050 | 0.0025 | 0.00127 |
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Rukavishnikov, V.A.; Rukavishnikova, E.I. Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary. Symmetry 2019, 11, 1455. https://doi.org/10.3390/sym11121455
Rukavishnikov VA, Rukavishnikova EI. Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary. Symmetry. 2019; 11(12):1455. https://doi.org/10.3390/sym11121455
Chicago/Turabian StyleRukavishnikov, Viktor A., and Elena I. Rukavishnikova. 2019. "Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary" Symmetry 11, no. 12: 1455. https://doi.org/10.3390/sym11121455
APA StyleRukavishnikov, V. A., & Rukavishnikova, E. I. (2019). Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary. Symmetry, 11(12), 1455. https://doi.org/10.3390/sym11121455