The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity
Mathematical models of fracture physics and mechanics are boundary value problems for differential equations and systems of equations with a singularity. There are two classes of problems with a singularity: with coordinated and uncoordinated degeneracy of the input data, depending on the behavior o...
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MDPI AG
2023-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/15/3272 |
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| author | Viktor A. Rukavishnikov Elena I. Rukavishnikova |
| author_facet | Viktor A. Rukavishnikov Elena I. Rukavishnikova |
| author_sort | Viktor A. Rukavishnikov |
| collection | DOAJ |
| description | Mathematical models of fracture physics and mechanics are boundary value problems for differential equations and systems of equations with a singularity. There are two classes of problems with a singularity: with coordinated and uncoordinated degeneracy of the input data, depending on the behavior of the coefficients of the equation. Finite element methods with the first order of convergence rate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>h</mi><mo>)</mo></mrow></semantics></math></inline-formula> have been created to find an approximate solution to these problems. We construct a scheme of the weighted finite element method of high degree of accuracy for the boundary value problem with uncoordinated degeneracy of the input data and singularity of the solution. The rate of convergence of an approximate solution of the proposed finite element method to the exact <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mi>ν</mi></msub></semantics></math></inline-formula>-generalized solution in the weight set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>W</mi><mrow><mn>2</mn><mo>,</mo><mi>ν</mi><mo>+</mo><mfrac><mi>β</mi><mn>2</mn></mfrac><mo>+</mo><mn>2</mn></mrow><mn>1</mn></msubsup><mrow><mo>(</mo><mo>Ω</mo><mo>,</mo><mi>δ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is investigated. The estimation of finite element approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> is established. |
| first_indexed | 2024-03-11T00:22:35Z |
| format | Article |
| id | doaj.art-87f0d0584c184af883b892f2e44d5656 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T00:22:35Z |
| publishDate | 2023-07-01 |
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| series | Mathematics |
| spelling | doaj.art-87f0d0584c184af883b892f2e44d56562023-11-18T23:14:25ZengMDPI AGMathematics2227-73902023-07-011115327210.3390/math11153272The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with SingularityViktor A. Rukavishnikov0Elena I. Rukavishnikova1Computing Center of Far Eastern Branch Russian Academy of Sciences, Kim Yu Chen Str. 65, 680000 Khabarovsk, RussiaComputing Center of Far Eastern Branch Russian Academy of Sciences, Kim Yu Chen Str. 65, 680000 Khabarovsk, RussiaMathematical models of fracture physics and mechanics are boundary value problems for differential equations and systems of equations with a singularity. There are two classes of problems with a singularity: with coordinated and uncoordinated degeneracy of the input data, depending on the behavior of the coefficients of the equation. Finite element methods with the first order of convergence rate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>h</mi><mo>)</mo></mrow></semantics></math></inline-formula> have been created to find an approximate solution to these problems. We construct a scheme of the weighted finite element method of high degree of accuracy for the boundary value problem with uncoordinated degeneracy of the input data and singularity of the solution. The rate of convergence of an approximate solution of the proposed finite element method to the exact <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mi>ν</mi></msub></semantics></math></inline-formula>-generalized solution in the weight set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>W</mi><mrow><mn>2</mn><mo>,</mo><mi>ν</mi><mo>+</mo><mfrac><mi>β</mi><mn>2</mn></mfrac><mo>+</mo><mn>2</mn></mrow><mn>1</mn></msubsup><mrow><mo>(</mo><mo>Ω</mo><mo>,</mo><mi>δ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is investigated. The estimation of finite element approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> is established.https://www.mdpi.com/2227-7390/11/15/3272finite element method of high degree of accuracyboundary value problem with singularityRν-generalized solution |
| spellingShingle | Viktor A. Rukavishnikov Elena I. Rukavishnikova The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity Mathematics finite element method of high degree of accuracy boundary value problem with singularity Rν-generalized solution |
| title | The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity |
| title_full | The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity |
| title_fullStr | The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity |
| title_full_unstemmed | The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity |
| title_short | The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity |
| title_sort | finite element method of high degree of accuracy for boundary value problem with singularity |
| topic | finite element method of high degree of accuracy boundary value problem with singularity Rν-generalized solution |
| url | https://www.mdpi.com/2227-7390/11/15/3272 |
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