{"title":"Discontinuous Galerkin Two-Grid Method for the Transient Navier–Stokes Equations","authors":"Kallol Ray, Deepjyoti Goswami, Saumya Bajpai","doi":"10.1515/cmam-2023-0035","DOIUrl":null,"url":null,"abstract":"In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0035_ineq_0001.png\" /> <jats:tex-math>L^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-11-24","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-2023-0035","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and L2L^{2}-norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme.
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.