[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82160-en":3,"doc-seo-82160-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82160,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","A Linearized and Structure-Preserving Mixed Virtual Element Method for the Extended Fisher-Kolmogorov Equation","A linearized, structure-preserving numerical algorithm is developed for the extended Fisher–Kolmogorov (EFK) equation using leap-frog discretization in time and a mixed virtual element discretization in space. The scheme’s main results include a rigorous proof of the fully discrete energy dissipation property and an unconditionally optimal convergence analysis via inverse inequalities, with the key proof step based on classified analysis of the relationship between the time step τ and mesh size h. Numerical experiments validate both accuracy and energy behavior.","arXiv :2607 .09044v1 [math .NA] 10 Jul 2026  \nA Linearized and structure-preserving mixed virtual element method for the extended Fisher-Kolmogorov equation  \nZhen Guan∗, Xianxian Cao, Houchao Zhang, Junjun Wang  \nSchool of Mathematics and Statistics, Pingdingshan University, Pingdingshan, 467000, China  \nAbstract  \nIn thsi paper, based on the leap-frog discretization in time and the mixed virtual element discretization in space, we developed a linearized and structure-preserving numerical algorithm. The main contributions of this work lie in that we not only provide a rigorous proof of the energy dissipation property of the fully discrete numerical scheme, but also establish the unconditionally optimal convergence analysis by means of a inverse inequality. The core of the proof lies in the classified discussion of the relationship between τ and h. Finally, two numerical examples are provided to validate the correctness of the theoretical analysis as well as the energy dissipation property of the proposed scheme.  \nKeywords: Leap-frog, Mixed virtual element, Structure-preserving, Unconditionally optimal convergence, Energy dissipation  \n1. Introduction  \nIn this work, we develop a structure-preserving numerical scheme for solving the following extended Fisher-Kolmogorov equation (EFK) in polygonal mesh  \nut + γ∆2 u − ∆u + u3 − u = 0 , (x, t) ∈ Ω × (0, T], (1 . 1)  \nu = ∆u = 0 , (x, t) ∈ ∂Ω × (0, T], (1.2)  \nu (x, 0) = u0 (x), x ∈ Ω , (1.3)  \nwhere γ is a positive constant, u (x, t) and u0 (x) are real-valued functions, Ω ⊂ R2 is a bounded convex polygonal domain, ∂Ω is the boundary of Ω and ∆ is the Laplace operator.  \nThe EFK equation is derived by adding a fourth-order term to the standard Fisher-Kolmogorov (FK) equation, which constitutes an important class of nonlinear fourth-order evolution equations. It has important applications in population genetics [1], domain wall propagation in liquid crystals [2], and the growth process of primary brain tumors [3] . Due to the high cost of obtaining exact solutions to nonlinear partial differential equations, numerous effective numerical methods have been developed by scholars in recent years. For example, Danumjaya and Pani [4] developed a first-order fully discrete numerical algorithm by adopting the C 1-conforming finite element method and the implicit Euler scheme, and rigorously proved the convergence of the proposed numerical algorithm. Liu and Yin [5] derived a parameter-free discontinuous Galerkin algorithm by adopting scalar auxiliary variable time discretization for solving a class of fourth-order gradient flow problems. Boujlida et al. [6] proposed a three-layer  \n∗ Corresponding author.  \nEmail address: [zhenguan1993@foxmail.com](zhenguan1993@foxmail.com) (Zhen Guan)  \ncompact difference scheme for solving the one-dimensional EFK equation, and derived the unique solvability and convergence of the scheme via the energy analysis method. Kumar and Natara [7] adopted the Euler and Crank-Nicolson numerical schemes to develop a hybrid high-order discretization method for solving the nonlinear EFK and FK equations, and analyzed the corresponding temporal and spatial error estimates. Chauhan and Chaudhary [8] investigated the space-time isogeometric method for a class of linear fourth-order evolution problems. The core idea is to introduce a auxiliary variable to decompose the fourth-order problem into a system of second-order equations. For the extended Fisher–Kolmogorov equation with clamped boundary conditions, Das and Nataraj [9] performed spatial discretization via the lowest-order nonstandard finite element method and implemented temporal discretization using the backward Euler scheme. Yang et al. [10] constructed a Crank–Nicolson mixed Galerkin scheme for the two-dimensional EFK equation, and conducted theoretical analyses of its convergence and superconvergence errors. Abbaszadeh et al. [11] adopted the interpolating element-free Galerkin method to solve the nonlinear ","cbCaigTm1hyHA0Ux","https://ap.wps.com/l/cbCaigTm1hyHA0Ux","pdf",6263092,1,22,"English","en",105,"# Introduction\n## Numerical formulation of the extended Fisher–Kolmogorov equation\n## Motivation for linearized, structure-preserving schemes\n## Related work and discretization strategies","[{\"question\":\"What numerical method is proposed for the extended Fisher–Kolmogorov equation?\",\"answer\":\"The method combines leap-frog discretization in time with a mixed virtual element discretization in space, producing a linearized, structure-preserving fully discrete scheme.\"},{\"question\":\"How is energy dissipation guaranteed in the proposed scheme?\",\"answer\":\"A rigorous proof establishes the energy dissipation property for the fully discrete numerical scheme, showing it preserves the equation’s dissipative structure.\"},{\"question\":\"What ensures the convergence analysis is unconditionally optimal?\",\"answer\":\"Unconditionally optimal convergence is obtained using an inverse inequality approach, with the proof relying on a classified discussion of the relationship between the time step τ and the mesh size h.\"}]",1784178510,55,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-linearized-and-structure-preserving-mixed-virtual-element-method-for-the-extended-fisher-kolmogorov-equation","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-linearized-and-structure-preserving-mixed-virtual-element-method-for-the-extended-fisher-kolmogorov-equation/82160/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What numerical method is proposed for the extended Fisher–Kolmogorov equation?","Question",{"text":75,"@type":76},"The method combines leap-frog discretization in time with a mixed virtual element discretization in space, producing a linearized, structure-preserving fully discrete scheme.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is energy dissipation guaranteed in the proposed scheme?",{"text":80,"@type":76},"A rigorous proof establishes the energy dissipation property for the fully discrete numerical scheme, showing it preserves the equation’s dissipative structure.",{"name":82,"@type":73,"acceptedAnswer":83},"What ensures the convergence analysis is unconditionally optimal?",{"text":84,"@type":76},"Unconditionally optimal convergence is obtained using an inverse inequality approach, with the proof relying on a classified discussion of the relationship between the time step τ and the mesh size h.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]