A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion Theory
In this paper, we discuss a class of nonlocal parabolic systems with nonlinear boundary conditions arising from the thermal explosion theory. First, we prove the local existence and uniqueness of the classical solution using the Leray–Schauder fixed-point theorem. Then, we analyze three Galerkin app...
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2025-03-01
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| author | Qipeng Guo Yu Zhang Baoqiang Yan |
| author_facet | Qipeng Guo Yu Zhang Baoqiang Yan |
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| description | In this paper, we discuss a class of nonlocal parabolic systems with nonlinear boundary conditions arising from the thermal explosion theory. First, we prove the local existence and uniqueness of the classical solution using the Leray–Schauder fixed-point theorem. Then, we analyze three Galerkin approximations of the system and derive the optimal-order error estimates: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>h</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> norm for continuous-time Galerkin approximation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>h</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mo>Δ</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> norm for Crank–Nicolson Galerkin approximation, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>h</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mo>Δ</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> norms for extrapolated Crank–Nicolson Galerkin approximation. |
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| language | English |
| publishDate | 2025-03-01 |
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| spelling | doaj-art-92a99bbea41c445fa1e4588c9f4fb6c72025-08-20T02:59:00ZengMDPI AGMathematics2227-73902025-03-0113586110.3390/math13050861A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion TheoryQipeng Guo0Yu Zhang1Baoqiang Yan2School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, ChinaSchool of Mathematics and Statistics, Shandong Normal University, Jinan 250358, ChinaSchool of Mathematics and Statistics, Shandong Normal University, Jinan 250358, ChinaIn this paper, we discuss a class of nonlocal parabolic systems with nonlinear boundary conditions arising from the thermal explosion theory. First, we prove the local existence and uniqueness of the classical solution using the Leray–Schauder fixed-point theorem. Then, we analyze three Galerkin approximations of the system and derive the optimal-order error estimates: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>h</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> norm for continuous-time Galerkin approximation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>h</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mo>Δ</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> norm for Crank–Nicolson Galerkin approximation, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>h</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mo>Δ</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> norms for extrapolated Crank–Nicolson Galerkin approximation.https://www.mdpi.com/2227-7390/13/5/861Galerkin finite element methodnonlocal parabolic systemfixed-point theoremnonlinear boundary conditionsuniquenesserror estimate |
| spellingShingle | Qipeng Guo Yu Zhang Baoqiang Yan A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion Theory Mathematics Galerkin finite element method nonlocal parabolic system fixed-point theorem nonlinear boundary conditions uniqueness error estimate |
| title | A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion Theory |
| title_full | A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion Theory |
| title_fullStr | A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion Theory |
| title_full_unstemmed | A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion Theory |
| title_short | A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion Theory |
| title_sort | galerkin finite element method for a nonlocal parabolic system with nonlinear boundary conditions arising from the thermal explosion theory |
| topic | Galerkin finite element method nonlocal parabolic system fixed-point theorem nonlinear boundary conditions uniqueness error estimate |
| url | https://www.mdpi.com/2227-7390/13/5/861 |
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