{"title":"一类三阶对流扩散型奇摄动问题的局部不连续伽辽金方法","authors":"Li Yan, Zhoufeng Wang, Yao Cheng","doi":"10.1515/cmam-2022-0176","DOIUrl":null,"url":null,"abstract":"The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>O</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msup> <m:mi>N</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2022-0176_eq_0280.png\" /> <jats:tex-math>{O(N^{-(k+\\frac{1}{2})})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2022-0176_eq_0414.png\" /> <jats:tex-math>{k\\geq 0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the maximum degree of piecewise polynomials used in discrete space, and <jats:italic>N</jats:italic> is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov–Shishkin-type, and Bakhvalov-type. Numerical experiments are conducted to test the theoretical results.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Local Discontinuous Galerkin Method for a Third-Order Singularly Perturbed Problem of Convection-Diffusion Type\",\"authors\":\"Li Yan, Zhoufeng Wang, Yao Cheng\",\"doi\":\"10.1515/cmam-2022-0176\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>O</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:msup> <m:mi>N</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2022-0176_eq_0280.png\\\" /> <jats:tex-math>{O(N^{-(k+\\\\frac{1}{2})})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2022-0176_eq_0414.png\\\" /> <jats:tex-math>{k\\\\geq 0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the maximum degree of piecewise polynomials used in discrete space, and <jats:italic>N</jats:italic> is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov–Shishkin-type, and Bakhvalov-type. Numerical experiments are conducted to test the theoretical results.\",\"PeriodicalId\":48751,\"journal\":{\"name\":\"Computational Methods in Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2022-12-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Methods in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/cmam-2022-0176\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/cmam-2022-0176","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Local Discontinuous Galerkin Method for a Third-Order Singularly Perturbed Problem of Convection-Diffusion Type
The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost O(N-(k+12)){O(N^{-(k+\frac{1}{2})})} (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, k≥0{k\geq 0} is the maximum degree of piecewise polynomials used in discrete space, and N is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov–Shishkin-type, and Bakhvalov-type. Numerical experiments are conducted to test the theoretical results.
期刊介绍:
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.
CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.
The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.