[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82237-en":3,"doc-seo-82237-105":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},82237,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Optimal Finite Element Error Estimates and Newton Convergence for a Quasilinear Elliptic Problem","Studies finite element approximations of a quasilinear elliptic heat-conduction problem under inhomogeneous mixed boundary conditions. The temperature- and position-dependent conductivity tensor is matrix-valued, anisotropic, and may be nonsymmetric, making the operator nonlinear, nonmonotone, and nonpotential. The Galerkin nonlinear algebraic system is solved by Newton’s method, supported by a computable Newton–Kantorovich criterion and a mesh-dependent stopping rule that makes algebraic error asymptotically negligible. Every discrete solution achieves optimal rates O(h) in H1 and O(h2) in L2, confirmed by 2D–3D numerics.","arXiv :2607 .09181v1 [math .NA] 10 Jul 2026  \nOptimal finite element error estimates and Newton convergence for aquasilinear elliptic problem with mixed boundary conditions  \nMihai Bucataru 1,2,∗  \n1 Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest,  \n14 Academiei, 010014 Bucharest, Romania  \n2“Gheorghe Mihoc – Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 13 Calea 13 Septembrie, 050711 Bucharest, Romania  \nAbstract  \nThe article studies finite element approximations of a quasilinear elliptic heat-conduction problem withinhomogeneous mixed boundary conditions. The conductivity tensor is matrix-valued, anisotropic, possibly nonsymmetric, and dependent on both position and temperature, rendering the problem nonlinear, nonmonotone, and nonpotential. The nonlinear algebraic system arising from the Galerkin discretization is solved using Newton’s method, with a posteriori guarantees provided by a computable Newton–Kantorovich criterion and a mesh-dependent stopping rule that ensures that the algebraic error is asymptotically negligible relative to the discretization error. Since the discrete solution need not be unique, we prove that every discrete solution satisfies the optimal convergence rates O (h) in the H 1-norm and O (h2 ) in the L2-norm. The analysis combines mixed-boundary elliptic regularity with an Aubin–Nitsche duality argument adapted to the quasilinear setting. Numerical experiments in two and three dimensions confirm the predicted convergence rates and demonstrate the feasibility of the proposed criterion.  \n1 Introduction  \nQuasilinear elliptic problems with solution-dependent anisotropic diffusion arise naturally in models of steady-state heat conduction in nonlinear inhomogeneous media, where the material response may vary with both position and temperature. The finite element approximation of nonlinear elliptic boundary value problems has been extensively studied, especially when the associated nonlinear operator is strongly monotone and Lipschitz continuous. In such a framework, an analogue of C´ea’s lemma is available and optimal H 1-error estimates for Lagrange finite elements can be derived by rather standard arguments; see, for instance,[8, 14, 15, 27, 38] . The problem considered in the present paper belongs, however, to a more delicate class. The diffusion tensor is matrix-valued, anisotropic, possibly nonsymmetric, and temperature-dependent, so the corresponding operator is in general neither monotone nor potential. Moreover, for d > 1, the usual Kirchhoff transformation (see [27]) cannot reduce the equation to a linear problem, even when the conductivity is independent of the spatial variable, because the conductivity is a matrix rather than a scalar function. This structural feature makes both the theoretical and numerical analysis substantially more involved. One-dimensional examples illustrating the nonmonotone and nonpotential character of this class of problems were discussed in [26] .  \nSeveral analytical results are known for quasilinear elliptic problems of this type. Existence results for weak solutions, under various boundary conditions, were obtained using compactness and weak-continuity methods in [16, 33], while uniqueness and comparison results for related nonpotential and nonmonotone problems were investigated in [25, 24] . In particular, the Galerkin approximation of a quasilinear nonpotential elliptic problem of nonmonotone type was studied in [26], where existence of discrete solutions  \n∗ E-mail: [mihai.bucataru@fmi.unibuc.ro](mihai.bucataru@fmi.unibuc.ro) ; ORCID ID: 0000-0002-6503-2084  \nwas obtained by Brouwer’s fixed-point theorem and convergence of Galerkin approximations was proved, although without deriving convergence rates. The uniqueness of the discrete solution is a delicate issue in this setting. Indeed, early works already provided only restrictive sufficient conditions for ","cbCaikBndM4okj1K","https://ap.wps.com/l/cbCaikBndM4okj1K","pdf",3891326,2,1,23,"English","en",105,"# Introduction\n## Problem setting and motivation\n## Known theory and uniqueness issues\n## Related numerical methods and extensions","[{\"question\":\"What mathematical problem does the article analyze?\",\"answer\":\"It analyzes finite element approximations of a quasilinear elliptic heat-conduction problem with inhomogeneous mixed boundary conditions, where the conductivity depends on both position and temperature.\"},{\"question\":\"How is the discrete nonlinear system solved and controlled?\",\"answer\":\"The Galerkin discretization yields a nonlinear algebraic system solved by Newton’s method, with a computable Newton–Kantorovich criterion and a mesh-dependent stopping rule to keep algebraic error negligible.\"},{\"question\":\"What convergence rates are proved for discrete solutions?\",\"answer\":\"Every discrete solution satisfies optimal convergence rates O(h) in the H1 norm and O(h2) in the L2 norm, even without guaranteeing uniqueness of the discrete solution.\"}]",1784179046,58,{"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},"optimal-finite-element-error-estimates-and-newton-convergence-for-a-quasilinear-elliptic-problem","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/optimal-finite-element-error-estimates-and-newton-convergence-for-a-quasilinear-elliptic-problem/82237/",4,{"url":51,"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","2026-07-22","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 mathematical problem does the article analyze?","Question",{"text":75,"@type":76},"It analyzes finite element approximations of a quasilinear elliptic heat-conduction problem with inhomogeneous mixed boundary conditions, where the conductivity depends on both position and temperature.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the discrete nonlinear system solved and controlled?",{"text":80,"@type":76},"The Galerkin discretization yields a nonlinear algebraic system solved by Newton’s method, with a computable Newton–Kantorovich criterion and a mesh-dependent stopping rule to keep algebraic error negligible.",{"name":82,"@type":73,"acceptedAnswer":83},"What convergence rates are proved for discrete solutions?",{"text":84,"@type":76},"Every discrete solution satisfies optimal convergence rates O(h) in the H1 norm and O(h2) in the L2 norm, even without guaranteeing uniqueness of the discrete 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