[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83402-en":3,"doc-seo-83402-105":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},83402,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Unconstrained Scheme for Geometrically Constrained Gradient Flows","This paper develops an approximation scheme for gradient flows of harmonic maps, used as model problems in micromagnetics, liquid crystals, and nonlinear plate bending. Harmonic maps are critical points of the Dirichlet energy under the pointwise unit-length constraint. Existing time-stepping methods enforce the constraint by linearizing it each step, often requiring difficult degenerate saddle-point solves. The proposed method computes an unconstrained increment and then projects it pointwise onto the tangent space, achieving energy stability under mild step-size restrictions and higher computational efficiency. It includes a posteriori stability criteria and variable time-stepping, plus an extension to plate bending.","arXiv :2607 .08577v1 [math .NA] 9 Jul 2026  \nUNCONSTRAINED SCHEME FOR GEOMETRICALLY CONSTRAINED  \nGRADIENT FLOWS  \nS¨OREN BARTELS, LUCAS BOUCK, AND CHRISTIAN PALUS  \nAbstract. In this paper, we study the approximation of gradient flows of harmonic maps, which serve as model problems for applications in micromagnetics, liquid crystals, and nonlinear plate bending. Harmonic maps are vector fields that are critical points of the Dirichlet energy subject to the constraint that the vector field be unit length pointwise. Most existing time-stepping schemes for gradient flows deal with the constraint bylinearizing the unit length constraint at every step, which involves solving for the solution increment in the tangent space of the constraint. These schemes lead to robust control over the violation of the constraint, but require solving degenerate saddle point systems at every step that may be difficult to precondition. In this paper, we propose a scheme that first computes the unconstrained increment and then projects this increment pointwise onto the tangent space. With an additional stabilization, this scheme is energy stable under mild step size restrictions and provides robust control of the unit length constraint violation. Our new scheme only requires the solution of decoupled symmetric positive definite systems at every step, which translates to a large increase in computational efficiency. We also propose a computable a posteriori criterion and a variable time-stepping procedure that guarantee the stability of the scheme. We conclude with computational examples demonstrating the efficacy of the scheme, and present a computational extension of the scheme to nonlinear plate bending.  \n1. Introduction  \nMinimization problems and gradient flows with geometric or other nonlinear pointwise constraints arise in numerous settings such as liquid crystals [17], micromagnetics [19], and plate bending [18] . Beginning with Alouges [2], a successful approach to discretizing such equations linearizes the geometric constraint at every step of the gradient flow by restricting the discrete velocity to live in the tangent space to the constraint. After this step, the predicted solution no longer satisfies the nonlinear geometric constraint, and the algorithm in [2] projects the solution back onto the constraint. This projection step is stable at the continuous level for the Dirichlet energy, but it may lead to a lack of energy stability when discretizing with finite elements as explored in the work by the first author  \nDepartment of Applied Mathematics, University of Freiburg, Hermann–Herder–Str. 10, 79104 Freiburg, Germany. Email: [bartels@mathematik.uni-freiburg.de](bartels@mathematik.uni-freiburg.de).  \nDepartment of Mathematical Sciences, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA. Email: [lbouck@andrew.cmu.edu](lbouck@andrew.cmu.edu).  \nDepartment of Applied Mathematics, University of Freiburg, Hermann–Herder–Str. 10, 79104 Freiburg, Germany. Email: [christian.palus@mathematik.uni-freiburg.de](christian.palus@mathematik.uni-freiburg.de).  \nUNCONSTRAINED GRADIENT FLOW 2  \nin [4] . Additionally, it was proved in [4] that a weakly acute mesh is a sufficient condition for energy stability of the nonlinear projection step.  \nTo bypass energy stability issues with the nonlinear projection of [2], the work of the first author in [7] proposed foregoing the nonlinear projection step entirely, hence the name“projection-free scheme.” While the solution no longer satisfies the geometric constraint exactly, the energy stability of the scheme allows [7] to prove an error estimate of the constraint violation in terms of the step size. When the solution is smoother, one can prove error estimates of the projection-free scheme [8] . Projection-free schemes have also been extended to energy minimization methods in liquid crystals [23, 11], bending isometries [5], bilayer plates [9], and accelerated schemes for such problems [15","cbCaikH50aFVh0Q9","https://ap.wps.com/l/cbCaikH50aFVh0Q9","pdf",9530810,3,1,26,"English","en",105,"# Abstract\n# Introduction\n## Background and existing projection and projection-free schemes\n## Limitations: saddle-point systems and stability concerns\n## Proposed unconstrained increment with tangent-space projection\n## Main contributions and paper outline","[{\"question\":\"What problem does the paper study?\",\"answer\":\"The paper studies approximating gradient flows of harmonic maps with a pointwise unit-length geometric constraint, motivated by applications in micromagnetics, liquid crystals, and nonlinear plate bending.\"},{\"question\":\"How does the proposed scheme differ from existing time-stepping methods?\",\"answer\":\"Instead of enforcing the constraint by linearizing it at every step and solving for the increment in the tangent space directly, the scheme first computes an unconstrained increment and then projects it pointwise onto the tangent space, with added stabilization for stability.\"},{\"question\":\"What computational advantage does the new scheme provide?\",\"answer\":\"The method avoids solving degenerate saddle-point systems at every step and instead requires only decoupled symmetric positive definite systems, yielding substantial speed-ups in the reported computational 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problem does the paper study?","Question",{"text":75,"@type":76},"The paper studies approximating gradient flows of harmonic maps with a pointwise unit-length geometric constraint, motivated by applications in micromagnetics, liquid crystals, and nonlinear plate bending.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed scheme differ from existing time-stepping methods?",{"text":80,"@type":76},"Instead of enforcing the constraint by linearizing it at every step and solving for the increment in the tangent space directly, the scheme first computes an unconstrained increment and then projects it pointwise onto the tangent space, with added stabilization for stability.",{"name":82,"@type":73,"acceptedAnswer":83},"What computational advantage does the new scheme provide?",{"text":84,"@type":76},"The method avoids solving degenerate saddle-point systems at every step and instead requires only decoupled symmetric positive definite systems, yielding substantial speed-ups in the reported computational examples.","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":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 & 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