[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84980-en":3,"doc-seo-84980-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84980,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Iterative Finite Element Approximation of Non-Monotone Semilinear Diffusion-Reaction Equations","Iterative linearized finite element methods are developed for semilinear elliptic boundary value problems with reaction terms of asymptotically linear growth, without imposing monotonicity. The work combines a stabilized implicit-explicit (IMEX) iteration with first-order finite elements on graded meshes tailored to polygonal corner singularities. Corner-regularity is handled via estimates in corner-weighted Sobolev spaces. A non-standard analysis of discrete Galerkin limits from the iterative process yields optimal a priori convergence rates, supported by numerical experiments.","arXiv :2607 .07327v1 [math .NA] 8 Jul 2026  \nITERATIVE FINITE ELEMENT APPROXIMATION OF NON-MONOTONE  \nSEMILINEAR DIFFUSION-REACTION EQUATIONS  \nFLORIAN SPICHER AND THOMAS P. WIHLER  \nAbstract. We study iterative linearized finite element methods for the numerical approximation of semilinear elliptic boundary value problems with nonlinear reaction terms of asymptotically linear growth. Our approach reaches considerably beyond the classical theory by allowing for nonlinearities that are not necessarily monotone. We investigate a stabilized implicit-explicit (IMEX) iteration combined with first-order finite element discretizations on graded meshes that are able to resolve corner singularities in polygonal domains. To account for the reduced regularity of the analytical solutions, we establish regularity estimates in corner-weighted Sobolev spaces. The principal novelty of the paper is a non-standard analysis of the approximation properties of the discrete Galerkin limits resulting from the iterative process, which yields optimal a  \npriori convergence estimates. Numerical experiments underline the theoretical findings.  \n1. Introduction  \nOn a bounded polygonal domain Ω ⊂ R2 , we seek numerical approximations of solutions u ∈ H10(Ω) of the semilinear elliptic boundary value problem  \n−∆u = f (·, u) in Ω (1.1a)  \nu = 0 on ∂Ω . (1.1b)  \nThe focus of this paper is on nonlinear reaction terms f : Ω × R → R for which no monotonicity assumptions are imposed. This implicates substantial challenges for the numerical analysis of the underlying problem, as standard approximation techniques (such as C´ea’s lemma in the finite element setting) or contractive iterative schemes (based on Banach’s fixed-point theorem) are no longer readily applicable; for a priori error estimates in the monotone case, we refer to the recent works [HHSW26, SW25, Vex25] . Nevertheless, if f exhibits asymptotically linear growth in the second argument in the sense that  \n|f(·, s)| ≤ C(1 + |s|) as |s| → ∞ [a.e. in](a.e. in) Ω, (1.2)  \nthen the introduction of a suitable L2-stabilization into (1 . 1a) restores some key monotonicity properties. This crucial observation can also be exploited for numerical purposes: in fact, for a suitable initial guess, and a sufficiently small (fixed) parameter ∆t > 0, it can be shown that the iterative procedure given by  \n−∆un+1 + tun+1 = tun + f(·, un ) in Ω (1.3a)  \nun+1 = 0 on ∂Ω, (1 .3b)  \nfor n ≥ 0, generates a sequence {un }n≥0 that converges weakly to a solution u ∈ H10(Ω) of (1.1a); we refer the interested reader to the method of sub-and supersolutions in the elliptic PDE literature  \nMathematics Institute, University of Bern, CH-3012 Switzerland  \n2020 Mathematics Subject Classification. 47J25, 65J15, 65N30 .  \nKey words and phrases. Semilinear elliptic boundary value problems, non-monotone problems, corner-weighted Sobolev spaces, elliptic corner singularities in polygons, finite element methods, optimal convergence, graded meshes, Aubin-Nitsche trick.  \nThe authors acknowledge the financial support of the Swiss National Science Foundation (SNSF), Grant No. 200021 212868.  \n2 ITERATIVE FEM FOR NON-MONOTONE SEMILINEAR ELLIPTIC PDES  \n(see, e.g.,[Eva10, §9.3]) . Alternatively, the above iteration can be interpreted as an implicit-explicit (IMEX) time-discretization scheme of the semilinear parabolic evolution equation  \n∂tv (x, t) − ∆v(x, t) = f(x, v(x, t)) (x, t) ∈ Ω × (0 , ∞ ) ,  \nfor a time step ∆t > 0.  \nFrom a practical point of view, it is important to note that (1.3) amounts to a linear solve at each iteration step, making the resulting procedure particularly appealing for numerical approximation once it is discretized in space, for instance, by finite element methods. Indeed, in the recent work [AHW23], under uniformly bounded derivative conditions on the reaction term f (as in (1.2) and to be specified later on in §2.2), the stabilized IMEX iteration (1.3) has been shown to yield (weak) convergence in closed subspa","cbCairUoRlFKUmfm","https://ap.wps.com/l/cbCairUoRlFKUmfm","pdf",856880,4,1,25,"English","en",105,"# Introduction\n## Problem setting and iteration concept\n# Iterative FEM for non-monotone semilinear elliptic PDEs\n## Challenges without monotonicity\n## Stabilized IMEX scheme and energy-based framework\n## Regularity in corner-weighted Sobolev spaces\n## Contributions and convergence goals","[{\"question\":\"What kind of PDEs are studied in this paper?\",\"answer\":\"The paper considers semilinear elliptic boundary value problems on bounded polygonal domains with reaction terms that grow asymptotically linearly and without monotonicity assumptions.\"},{\"question\":\"How does the method handle non-monotone nonlinearities?\",\"answer\":\"It introduces an L2-stabilization that restores key monotonicity properties, and uses a stabilized implicit-explicit (IMEX) iteration at each step.\"},{\"question\":\"Why are graded meshes and corner-weighted Sobolev spaces used?\",\"answer\":\"Because solutions may exhibit elliptic corner singularities at polygon vertices; the analysis establishes regularity in corner-weighted Sobolev spaces, which motivates graded refinement near corners.\"}]",1784199977,63,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"iterative-finite-element-approximation-of-non-monotone-semilinear-diffusion-reaction-equations","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/iterative-finite-element-approximation-of-non-monotone-semilinear-diffusion-reaction-equations/84980/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","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 kind of PDEs are studied in this paper?","Question",{"text":75,"@type":76},"The paper considers semilinear elliptic boundary value problems on bounded polygonal domains with reaction terms that grow asymptotically linearly and without monotonicity assumptions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method handle non-monotone nonlinearities?",{"text":80,"@type":76},"It introduces an L2-stabilization that restores key monotonicity properties, and uses a stabilized implicit-explicit (IMEX) iteration at each step.",{"name":82,"@type":73,"acceptedAnswer":83},"Why are graded meshes and corner-weighted Sobolev spaces used?",{"text":84,"@type":76},"Because solutions may exhibit elliptic corner singularities at polygon vertices; the analysis establishes regularity in corner-weighted Sobolev spaces, which motivates graded refinement near corners.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]