An hp-version of the discontinuous Galerkin method for fractional integro-differential equations with weakly singular kernels

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Yanping Chen, Zhenrong Chen, Yunqing Huang
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引用次数: 0

Abstract

This paper suggests an hp-discontinuous Galerkin approach for the fractional integro-differential equations with weakly singular kernels. The key idea behind our method is to first convert the fractional integro-differential equations into the second kind of Volterra integral equations, and then solve the equivalent integral equations using the hp-discontinuous Galerkin method. We establish prior error bounds in the \(L^{2}\)-norm that is entirely explicit about the local mesh sizes, local polynomial degrees, and local regularities of the exact solutions. The use of geometrically refined meshes and linearly increasing approximation orders demonstrates, in particular, that exponential convergence is achievable for solutions with endpoint singularities. Numerical results indicate the usefulness of the proposed method.

Abstract Image

弱奇异内核分式积分微分方程的非连续伽勒金方法 hp 版本
本文针对具有弱奇异内核的分数积分微分方程提出了一种 hp-非连续 Galerkin 方法。我们方法的主要思想是首先将分式微分方程转换为第二类 Volterra 积分方程,然后使用 hp-非连续 Galerkin 方法求解等效积分方程。我们建立了 \(L^{2}\)-norm 的先验误差边界,完全明确了精确解的局部网格尺寸、局部多项式度和局部正则性。几何细化网格和线性递增近似阶数的使用特别表明,对于具有端点奇异性的解,指数收敛是可以实现的。数值结果表明了所提方法的实用性。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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