{"title":"非线性耦合时分数薛定谔方程线性化变换 L1 虚拟元素法的无条件误差分析","authors":"","doi":"10.1016/j.cam.2024.116283","DOIUrl":null,"url":null,"abstract":"<div><div>This paper constructs a linearized transformed <span><math><mrow><mi>L</mi><mn>1</mn></mrow></math></span> virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical <span><math><mi>s</mi></math></span>-fractional differential system derived from a smoothing transformation of variables <span><math><mrow><mi>t</mi><mo>=</mo><msup><mrow><mi>s</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>α</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></math></span>. By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Unconditional error analysis of the linearized transformed L1 virtual element method for nonlinear coupled time-fractional Schrödinger equations\",\"authors\":\"\",\"doi\":\"10.1016/j.cam.2024.116283\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This paper constructs a linearized transformed <span><math><mrow><mi>L</mi><mn>1</mn></mrow></math></span> virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical <span><math><mi>s</mi></math></span>-fractional differential system derived from a smoothing transformation of variables <span><math><mrow><mi>t</mi><mo>=</mo><msup><mrow><mi>s</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>α</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></math></span>. By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.</div></div>\",\"PeriodicalId\":50226,\"journal\":{\"name\":\"Journal of Computational and Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Computational and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0377042724005272\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042724005272","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
本文为广义非线性耦合时间分数薛定谔方程构建了线性化变换 L1 虚拟元素方法。此类问题的解通常在开始时表现出奇异行为。为了避免这一缺陷,我们引入了一个由变量 t=s1/α, 0<α<1 的平滑变换导出的相同 s 分式微分方程系统。此外,我们还在不限制网格比的情况下,以 L2 规范推导出了所提出的完全离散方案的无条件最优误差边界。最后,对一组多边形网格进行了数值测试,以验证理论结果。
Unconditional error analysis of the linearized transformed L1 virtual element method for nonlinear coupled time-fractional Schrödinger equations
This paper constructs a linearized transformed virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical -fractional differential system derived from a smoothing transformation of variables , . By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in -norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.
期刊介绍:
The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest.
The audience consists of: applied mathematicians, numerical analysts, computational scientists and engineers.