{"title":"一类非线性四阶波动方程的保能混合有限元法","authors":"Mingyan He , Fengchi Tu , Jia Tian , Pengtao Sun","doi":"10.1016/j.cam.2025.116990","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, a type of energy-preserving mixed finite element method (FEM) is proposed for a class of generalized nonlinear fourth-order wave equations by means of the Raviart–Thomas mixed element and Crank–Nicolson temporal discretization in its full discretization, where besides the primary variable, its numerical gradient and Laplacian are also obtained, simultaneously, with the energy conservation property and optimal convergence rates in their energy norms that are rigorously proved in the theoretical analyses. In addition, a particular projection technique is introduced and analyzed in a coupling and mixed finite element form to assist in proving the optimal convergence properties. Numerical experiments are carried out to validate all theoretical results. The developed numerical method provides a guide to solve physical vibration problems of beams and thin plates, which own the form of nonlinear fourth-order wave equations, stably and accurately for a long time in a way that ensures energy conservation.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"474 ","pages":"Article 116990"},"PeriodicalIF":2.6000,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An energy-preserving mixed finite element method for a class of nonlinear fourth-order wave equations\",\"authors\":\"Mingyan He , Fengchi Tu , Jia Tian , Pengtao Sun\",\"doi\":\"10.1016/j.cam.2025.116990\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper, a type of energy-preserving mixed finite element method (FEM) is proposed for a class of generalized nonlinear fourth-order wave equations by means of the Raviart–Thomas mixed element and Crank–Nicolson temporal discretization in its full discretization, where besides the primary variable, its numerical gradient and Laplacian are also obtained, simultaneously, with the energy conservation property and optimal convergence rates in their energy norms that are rigorously proved in the theoretical analyses. In addition, a particular projection technique is introduced and analyzed in a coupling and mixed finite element form to assist in proving the optimal convergence properties. Numerical experiments are carried out to validate all theoretical results. The developed numerical method provides a guide to solve physical vibration problems of beams and thin plates, which own the form of nonlinear fourth-order wave equations, stably and accurately for a long time in a way that ensures energy conservation.</div></div>\",\"PeriodicalId\":50226,\"journal\":{\"name\":\"Journal of Computational and Applied Mathematics\",\"volume\":\"474 \",\"pages\":\"Article 116990\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2025-08-07\",\"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/S0377042725005047\",\"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/S0377042725005047","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
An energy-preserving mixed finite element method for a class of nonlinear fourth-order wave equations
In this paper, a type of energy-preserving mixed finite element method (FEM) is proposed for a class of generalized nonlinear fourth-order wave equations by means of the Raviart–Thomas mixed element and Crank–Nicolson temporal discretization in its full discretization, where besides the primary variable, its numerical gradient and Laplacian are also obtained, simultaneously, with the energy conservation property and optimal convergence rates in their energy norms that are rigorously proved in the theoretical analyses. In addition, a particular projection technique is introduced and analyzed in a coupling and mixed finite element form to assist in proving the optimal convergence properties. Numerical experiments are carried out to validate all theoretical results. The developed numerical method provides a guide to solve physical vibration problems of beams and thin plates, which own the form of nonlinear fourth-order wave equations, stably and accurately for a long time in a way that ensures energy conservation.
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.