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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-03-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/5/861 |
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| Summary: | 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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| ISSN: | 2227-7390 |