| 时间分数阶次扩散方程的多层扩充算法 |
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| 引用本文： | 陈剑,曾泰山.时间分数阶次扩散方程的多层扩充算法[J].华南师范大学学报(自然科学版),2020,52(3):106-110. |
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| 作者姓名： | 陈剑 曾泰山 |
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| 作者单位： | 1.佛山科学技术学院数学与大数据学院, 佛山 528000 |
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| 基金项目： | 广东省自然科学基金;广东省普通高等学校特色创新类项目;国家自然科学基金 |
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| 摘 要： | 基于L1公式和多尺度Galerkin方法, 对具有α阶Caputo导数的时间分数阶次扩散方程建立了全离散格式；证明了全离散格式存在唯一解和具有最优收敛阶O(hr+τ2-α), r为分片多项式的次数；在每个时间层，对全离散格式所得线性方程组, 设计了多层扩充算法进行高效求解, 并保持着最优收敛阶；最后, 给出数值算例来验证理论分析的正确性.
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| 关 键 词： | L1逼近 多尺度正交基 多层扩充法 分数阶次扩散方程 收敛阶 |
| 收稿时间： | 2020-01-14 |
The Multilevel Augmentation Method for Solving Time Fractional Subdiffusion Equation |
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| Institution: | 1.School of Mathematics and Big Data, Foshan University, Foshan 528000, China2.School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China |
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| Abstract: | Based on L1 formula and the multiscale Galerkin method, a fully-discrete scheme is proposed for solving time fractional subdiffusion equations with α order Caputo fractional derivative. The existence and uniqueness of the solution of the fully-discrete scheme are proved, and the optimal convergence order O(hr+ τ 2-α) is also deduced, where r is the order of piecewise polynomials. A multilevel augmentation method (MAM) is developed to solve the linear systems resulting from the fully-discrete scheme at each time step, and MAM preserves the optimal convergence order. A numerical experiment is presented at last to show the validity of the theoretical analysis. |
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