{"title":"用混合有限元法求解纯诺伊曼问题","authors":"M. I. Ivanov, I. A. Kremer, Yu. M. Laevsky","doi":"10.1134/s1995423922040048","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>This paper proposes a new method for the numerical solution of the pure Neumann problem for the diffusion equation in a mixed formulation. The method is based on the inclusion of a condition of unique solvability of the problem in one of the equations of the system with a subsequent decrease in its order by using a Lagrange multiplier. The unique solvability of the problem thus obtained and its equivalence to the original mixed formulation in a subspace are proved. The problem is approximated on the basis of a mixed finite element method. The unique solvability of the resulting system of saddle point linear algebraic equations is investigated. Theoretical results are illustrated by computational experiments.</p>","PeriodicalId":43697,"journal":{"name":"Numerical Analysis and Applications","volume":"18 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Solving the Pure Neumann Problem by a Mixed Finite Element Method\",\"authors\":\"M. I. Ivanov, I. A. Kremer, Yu. M. Laevsky\",\"doi\":\"10.1134/s1995423922040048\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p>This paper proposes a new method for the numerical solution of the pure Neumann problem for the diffusion equation in a mixed formulation. The method is based on the inclusion of a condition of unique solvability of the problem in one of the equations of the system with a subsequent decrease in its order by using a Lagrange multiplier. The unique solvability of the problem thus obtained and its equivalence to the original mixed formulation in a subspace are proved. The problem is approximated on the basis of a mixed finite element method. The unique solvability of the resulting system of saddle point linear algebraic equations is investigated. Theoretical results are illustrated by computational experiments.</p>\",\"PeriodicalId\":43697,\"journal\":{\"name\":\"Numerical Analysis and Applications\",\"volume\":\"18 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-12-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Numerical Analysis and Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1134/s1995423922040048\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Analysis and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1995423922040048","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Solving the Pure Neumann Problem by a Mixed Finite Element Method
Abstract
This paper proposes a new method for the numerical solution of the pure Neumann problem for the diffusion equation in a mixed formulation. The method is based on the inclusion of a condition of unique solvability of the problem in one of the equations of the system with a subsequent decrease in its order by using a Lagrange multiplier. The unique solvability of the problem thus obtained and its equivalence to the original mixed formulation in a subspace are proved. The problem is approximated on the basis of a mixed finite element method. The unique solvability of the resulting system of saddle point linear algebraic equations is investigated. Theoretical results are illustrated by computational experiments.
期刊介绍：
Numerical Analysis and Applications is the translation of Russian periodical Sibirskii Zhurnal Vychislitel’noi Matematiki (Siberian Journal of Numerical Mathematics) published by the Siberian Branch of the Russian Academy of Sciences Publishing House since 1998.
The aim of this journal is to demonstrate, in concentrated form, to the Russian and International Mathematical Community the latest and most important investigations of Siberian numerical mathematicians in various scientific and engineering fields.
The journal deals with the following topics: Theory and practice of computational methods, mathematical physics, and other applied fields; Mathematical models of elasticity theory, hydrodynamics, gas dynamics, and geophysics; Parallelizing of algorithms; Models and methods of bioinformatics.
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