# Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

- (a)
- $$f\in {L}_{2,-1-\alpha}\left(\mathsf{\Omega}\right),$$
- (b)
- ${a}_{kk}\left(x\right)$ ($k=1,2$) are differentiable functions on $\mathsf{\Omega}$, such that the inequalities$$|{a}_{kk}\left(x\right)|\le {C}_{1}{\rho}^{-2\alpha}\left(x\right),$$$$\left|\frac{\partial {a}_{kk}\left(x\right)}{\partial {x}_{1}}\right|,\left|\frac{\partial {a}_{kk}\left(x\right)}{\partial {x}_{2}}\right|\le {C}_{2}{\rho}^{-2\alpha -1}\left(x\right),$$$$\sum _{k=1}^{2}{a}_{kk}\left(x\right){\xi}_{k}^{2}\ge {C}_{3}{\rho}^{-2\alpha}\left(x\right)\sum _{k=1}^{2}{\xi}_{k}^{2},\phantom{\rule{1.em}{0ex}}x\in \mathsf{\Omega},\phantom{\rule{1.em}{0ex}}{C}_{3}>0,$$
- (c)
- the function $a\left(x\right)$ satisfies the inequalities$$0<a\left(x\right)\le {C}_{4}{\rho}^{-2\alpha -2}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in \mathsf{\Omega}.$$

**Remark**

**1.**

**Remark**

**2.**

## 3. The Scheme of the Finite Element Method

## 4. Numerical Experiments

`In`the second case h changes due to a decrease in the width of the border strip b at a fixed number n (Figure 2b).

## 5. Conclusions

- An approximate generalized solution of Equation (1) on grids with an appropriate compression of nodes (close) to the boundary of the domain converges to an exact solution with a speed $O\left({h}^{2}\right)$ in the norm of the space ${L}_{2}\left(\mathsf{\Omega}\right)$ and $O\left(h\right)$ in the norm of the space ${W}_{2,\alpha}^{1}\left(\mathsf{\Omega}\right)$ (see Table 2 and Table 5);
- 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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**Figure 2.**The graph of the error ${\parallel e\parallel}_{{W}_{2,\alpha}^{1}\left(\mathsf{\Omega}\right)}$ on the grids ${R}_{q}$ and ${R}_{c}$ as h changes in a logarithmic scale for Model Problem 2. For ${R}_{c}$ (

**a**) $b=const=1/64$, n is a variable; (

**b**) n is a constant, b is a variable.

Quasi-Uniform Mesh (${\mathit{R}}_{\mathit{q}}$) | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |

Number of nodes N | 10,849,474 | $e\ge 3\times {10}^{-6}$ | 0.00% | 69 | |

$1\times {10}^{-6}\le e<3\times {10}^{-6}$ | 89.37% | 9,696,362 | |||

$7\times {10}^{-7}\le e<1\times {10}^{-6}$ | 10.61% | 1,151,232 | |||

h | 0.00055 | $3\times {10}^{-7}\le e<7\times {10}^{-7}$ | 0.01% | 1196 | |

$1\times {10}^{-7}\le e<3\times {10}^{-7}$ | 0.00% | 372 | |||

$0\le e<1\times {10}^{-7}$ | 0.00% | 243 | |||

Refined Mesh (${\mathit{R}}_{\mathit{c}}$) | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |

N | 10,755,478 | $e\ge 3\times {10}^{-6}$ | 0.00% | 0 | |

Number of nodes in domain ${\mathsf{\Omega}}_{2}$ | 10,661,162 | $1\times {10}^{-6}\le e<3\times {10}^{-6}$ | 0.00% | 0 | |

$7\times {10}^{-7}\le e<1\times {10}^{-6}$ | 0.00% | 4 | |||

h | 0.000556 | $3\times {10}^{-7}\le e<7\times {10}^{-7}$ | 44.46% | 4,781,879 | |

n | 3 | $1\times {10}^{-7}\le e<3\times {10}^{-7}$ | 55.44% | 5,962,761 | |

b | 1/1024 | $0\le e<1\times {10}^{-7}$ | 0.10% | 10,834 | |

Refined Mesh (${\mathit{R}}_{\mathit{c}}$) | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |

N | 4,974,486 | $e\ge 3\times {10}^{-6}$ | 0.00% | 0 | |

Number of nodes in domain ${\mathsf{\Omega}}_{2}$ | 4,241,164 | $1\times {10}^{-6}\le e<3\times {10}^{-6}$ | 0.00% | 0 | |

$7\times {10}^{-7}\le e<1\times {10}^{-6}$ | 0.00% | 0 | |||

h | 0.00087 | $3\times {10}^{-7}\le e<7\times {10}^{-7}$ | 2.75% | 136,821 | |

n | 18 | $1\times {10}^{-7}\le e<3\times {10}^{-7}$ | 83.18% | 4,137,684 | |

b | 1/128 | $0\le e<1\times {10}^{-7}$ | 14.07% | 699,981 |

**Table 2.**The errors ${\parallel e\parallel}_{{L}_{2}\left(\mathsf{\Omega}\right)}$ and ${\parallel e\parallel}_{{W}_{2,\alpha}^{1}\left(\mathsf{\Omega}\right)}$ for meshes ${R}_{q}$ and ${R}_{c}$ for Model Problem 1.

Quasi-Uniform Mesh (${\mathit{R}}_{\mathit{q}}$) | Refined Mesh (${\mathit{R}}_{\mathit{c}}$), $\mathit{b}=1/128$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{h}$ | ${\parallel \mathit{e}\parallel}_{{\mathit{L}}_{\mathbf{2}}\left(\mathsf{\Omega}\right)}$ | $\mathit{\beta}$ | ${\parallel \mathit{e}\parallel}_{{\mathit{W}}_{\mathbf{2},\mathit{\alpha}}^{\mathbf{1}}\left(\mathsf{\Omega}\right)}$ | $\mathit{\beta}$ | $\mathit{h}$ | ${\parallel \mathit{e}\parallel}_{{\mathit{L}}_{\mathbf{2}}\left(\mathsf{\Omega}\right)}$ | $\mathit{\beta}$ | ${\parallel \mathit{e}\parallel}_{{\mathit{W}}_{\mathbf{2},\mathit{\alpha}}^{\mathbf{1}}\left(\mathsf{\Omega}\right)}$ | $\mathit{\beta}$ |

0.0022 | $5.40\times {10}^{-6}$ | $3.83\times {10}^{-3}$ | 0.0035 | $3.00\times {10}^{-6}$ | $5.11\times {10}^{-3}$ | ||||

1.68 | 1.85 | 4.21 | 2.06 | ||||||

0.0011 | $3.21\times {10}^{-6}$ | $2.22\times {10}^{-3}$ | 0.00169 | $7.12\times {10}^{-7}$ | $2.48\times {10}^{-3}$ | ||||

1.83 | 1.72 | 4.11 | 2.03 | ||||||

0.00055 | $1.75\times {10}^{-6}$ | $1.36\times {10}^{-3}$ | 0.00083 | $1.73\times {10}^{-7}$ | $1.23\times {10}^{-3}$ |

**Table 3.**The distribution of the error e on the grids ${R}_{q}$ and ${R}_{c}$ as h changes for Model Problem 1.

Specified Limited Error | Quasi-Uniform Mesh (${\mathit{R}}_{\mathit{q}}$) | ||

$e\ge 3\times {10}^{-6}$ | |||

$1\times {10}^{-6}\le e<3\times {10}^{-6}$ | |||

$7\times {10}^{-7}\le e<1\times {10}^{-6}$ | |||

$3\times {10}^{-7}\le e<7\times {10}^{-7}$ | |||

$1\times {10}^{-7}\le e<3\times {10}^{-7}$ | |||

$0\le e<1\times {10}^{-7}$ | |||

N | 338,449 | 1,355,498 | 5,427,739 |

h | 0.0031 | 0.0015 | 0.00078 |

Specified Limited Error | Refined Mesh (${\mathit{R}}_{\mathit{c}}$), $\mathit{b}=\mathbf{1}/\mathbf{128}$ | ||

$\mathit{n}=\mathbf{5}$ | $\mathit{n}=\mathbf{10}$ | $\mathit{n}=\mathbf{20}$ | |

$e\ge 3\times {10}^{-6}$ | |||

$1\times {10}^{-6}\le e<3\times {10}^{-6}$ | |||

$7\times {10}^{-7}\le e<1\times {10}^{-6}$ | |||

$3\times {10}^{-7}\le e<7\times {10}^{-7}$ | |||

$1\times {10}^{-7}\le e<3\times {10}^{-7}$ | |||

$0\le e<1\times {10}^{-7}$ | |||

N | 425,760 | 1,569,052 | 6,129,755 |

Number of nodes in domain ${\mathsf{\Omega}}_{2}$ | 386,628 | 1,375,684 | 5,200,079 |

h in domain ${\mathsf{\Omega}}_{2}$ | 0.0029 | 0.0015 | 0.00079 |

Quasi-Uniform Mesh (${\mathit{R}}_{\mathit{q}}$) | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |

N | 2,713,152 | $e\ge 5\times {10}^{-3}$ | 50.40% | 1,367,499 | |

$2\times {10}^{-3}\le e<5\times {10}^{-3}$ | 49.60% | 1,345,653 | |||

$1\times {10}^{-3}\le e<2\times {10}^{-3}$ | 0.00% | 0 | |||

h | 0.00055 | $7\times {10}^{-4}\le e<1\times {10}^{-3}$ | 0.00% | 0 | |

$4\times {10}^{-4}\le e<7\times {10}^{-4}$ | 0.00% | 0 | |||

Refined Mesh (${\mathit{R}}_{\mathit{c}}$) | Absolute Error Distribution | Specified Limited Error | Percent | Number of Nodes | |

N | 3,254,432 | $e\ge 5\times {10}^{-3}$ | 0.00% | 0 | |

Number of nodes in domain ${\mathsf{\Omega}}_{2}$ | 2,484,744 | $2\times {10}^{-3}\le e<5\times {10}^{-3}$ | 0.00% | 0 | |

h in domain ${\mathsf{\Omega}}_{2}$ | 0.0011 | $1\times {10}^{-3}\le e<2\times {10}^{-3}$ | 26.32% | 856,521 | |

n | 27 | $7\times {10}^{-4}\le e<1\times {10}^{-3}$ | 31.94% | 1,039,571 | |

b | $1/64$ | $4\times {10}^{-4}\le e<7\times {10}^{-4}$ | 41.74% | 1,358,340 |

**Table 5.**The error ${\parallel e\parallel}_{{W}_{2,\alpha}^{1}\left(\mathsf{\Omega}\right)}$ for meshes ${R}_{q}$ and ${R}_{c}$ for Model Problem 2.

Quasi-Uniform Mesh (${\mathit{R}}_{\mathit{q}}$) | Refined Mesh (${\mathit{R}}_{\mathit{c}}$), $\mathit{b}=1/64$ | |||||
---|---|---|---|---|---|---|

$\mathit{h}$ | ${\parallel \mathit{e}\parallel}_{{\mathit{W}}_{\mathbf{2},\mathit{\alpha}}^{\mathbf{1}}\left({\Omega}\right)}$ | $\mathbf{\beta}$ | $\mathit{n}$ | $\mathit{h}$in domain${\mathsf{\Omega}}_{\mathbf{2}}$ | ${\parallel \mathit{e}\parallel}_{{\mathit{W}}_{\mathbf{2},\mathbf{\alpha}}^{\mathbf{1}}\left(\mathsf{\Omega}\right)}$ | $\mathbf{\beta}$ |

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 |

**Table 6.**The distribution of the error e on the grids ${R}_{q}$ and ${R}_{c}$ as h changes for Model Problem 2.

Specified Limited Error | Quasi-Uniform Mesh (${\mathit{R}}_{\mathit{q}}$) | ||

$e\ge 5\times {10}^{-3}$ | |||

$2\times {10}^{-3}\le e<5\times {10}^{-3}$ | |||

$1\times {10}^{-3}\le e<2\times {10}^{-3}$ | |||

$7\times {10}^{-4}\le e<1\times {10}^{-3}$ | |||

$4\times {10}^{-4}\le e<7\times {10}^{-4}$ | |||

N | 168,670 | 677,704 | 2,713,152 |

h | 0.0044 | 0.0022 | 0.0011 |

Specified Limited Error | Refined Mesh (${\mathit{R}}_{\mathit{c}}$) | ||

$\mathit{n}=\mathbf{6}$ | $\mathit{n}=\mathbf{12}$ | $\mathit{n}=\mathbf{24}$ | |

$e\ge 5\times {10}^{-3}$ | |||

$2\times {10}^{-3}\le e<5\times {10}^{-3}$ | |||

$1\times {10}^{-3}\le e<2\times {10}^{-3}$ | |||

$7\times {10}^{-4}\le e<1\times {10}^{-3}$ | |||

$4\times {10}^{-4}\le e<7\times {10}^{-4}$ | |||

N | 166,751 | 643,498 | 2,567,442 |

Number of nodes in domain ${\mathsf{\Omega}}_{2}$ | 139,635 | 514,952 | 1,972,944 |

h in domain ${\mathsf{\Omega}}_{2}$ | 0.0050 | 0.0025 | 0.00127 |

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Rukavishnikov, 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