The Optimal <i>L</i><sup>2</sup>-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular Kernel
This paper proposes a weak Galerkin (WG) finite element method for solving a multi-dimensional evolution equation with a weakly singular kernel. The temporal discretization employs the backward Euler scheme, while the fractional integral term is approximated via a piecewise constant function method....
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2025-06-01
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| author | Haopan Zhou Jun Zhou Hongbin Chen |
| author_facet | Haopan Zhou Jun Zhou Hongbin Chen |
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| description | This paper proposes a weak Galerkin (WG) finite element method for solving a multi-dimensional evolution equation with a weakly singular kernel. The temporal discretization employs the backward Euler scheme, while the fractional integral term is approximated via a piecewise constant function method. A fully discrete scheme is constructed by integrating the WG finite element approach for spatial discretization. <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 stability and convergence analysis of the fully discrete scheme are rigorously established. Numerical experiments are conducted to validate the theoretical findings and demonstrate optimal convergence order in both spatial and temporal directions. The numerical results confirm that the proposed method achieves an accuracy of the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mfenced separators="" open="(" close=")"><mi>τ</mi><mo>+</mo><msup><mi>h</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mfenced></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> and <i>h</i> represent the time step and spatial mesh size, respectively. This work extends previous studies on one-dimensional problems to higher spatial dimensions, providing a robust framework for handling evolution equations with a weakly singular kernel. |
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| institution | Kabale University |
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| spelling | doaj-art-5a636bb6d0ae4e94b50f742d27752b202025-08-20T03:27:32ZengMDPI AGFractal and Fractional2504-31102025-06-019636810.3390/fractalfract9060368The Optimal <i>L</i><sup>2</sup>-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular KernelHaopan Zhou0Jun Zhou1Hongbin Chen2Bangor College, Central South University of Forestry and Technology, Changsha 410004, ChinaCollege of Computer Science and Mathematics, Central South University of Forestry and Technology, Changsha 410004, ChinaCollege of Computer Science and Mathematics, Central South University of Forestry and Technology, Changsha 410004, ChinaThis paper proposes a weak Galerkin (WG) finite element method for solving a multi-dimensional evolution equation with a weakly singular kernel. The temporal discretization employs the backward Euler scheme, while the fractional integral term is approximated via a piecewise constant function method. A fully discrete scheme is constructed by integrating the WG finite element approach for spatial discretization. <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 stability and convergence analysis of the fully discrete scheme are rigorously established. Numerical experiments are conducted to validate the theoretical findings and demonstrate optimal convergence order in both spatial and temporal directions. The numerical results confirm that the proposed method achieves an accuracy of the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mfenced separators="" open="(" close=")"><mi>τ</mi><mo>+</mo><msup><mi>h</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mfenced></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> and <i>h</i> represent the time step and spatial mesh size, respectively. This work extends previous studies on one-dimensional problems to higher spatial dimensions, providing a robust framework for handling evolution equations with a weakly singular kernel.https://www.mdpi.com/2504-3110/9/6/368evolution equationweakly singular kernelweak Galerkin finite element methodbackward Euler methodstability and convergence |
| spellingShingle | Haopan Zhou Jun Zhou Hongbin Chen The Optimal <i>L</i><sup>2</sup>-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular Kernel Fractal and Fractional evolution equation weakly singular kernel weak Galerkin finite element method backward Euler method stability and convergence |
| title | The Optimal <i>L</i><sup>2</sup>-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular Kernel |
| title_full | The Optimal <i>L</i><sup>2</sup>-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular Kernel |
| title_fullStr | The Optimal <i>L</i><sup>2</sup>-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular Kernel |
| title_full_unstemmed | The Optimal <i>L</i><sup>2</sup>-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular Kernel |
| title_short | The Optimal <i>L</i><sup>2</sup>-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular Kernel |
| title_sort | optimal i l i sup 2 sup norm error estimate of a weak galerkin finite element method for a multi dimensional evolution equation with a weakly singular kernel |
| topic | evolution equation weakly singular kernel weak Galerkin finite element method backward Euler method stability and convergence |
| url | https://www.mdpi.com/2504-3110/9/6/368 |
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