[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83333-en":3,"doc-seo-83333-105":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},83333,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Algorithm XXXX: Computation of Finite Element Degree-of-Freedom Transformation Matrices","Finite element operators become more compute-intensive at higher polynomial degree, making high-degree methods attractive on modern CPU and GPU hardware. Existing work ensures stable high-degree finite element bases, yet efficient, automated construction of the degree-of-freedom map for arbitrary elements lacks a universal solution. The presented algorithm builds DOF maps for any Ciarlet-type element using only the element definition and reference-cell properties, and is implemented in Basix (FEniCSx).","arXiv :2607 .08172v1 [math .NA] 9 Jul 2026  \nAlgorithm XXXX: Computation of finite element degree-of-freedom transformation matrices  \nMATTHEW W. SCROGGS, Advanced Research Computing Centre, University College London, United Kingdom GARTH N. WELLS, Department of Engineering, University of Cambridge, United Kingdom  \nThe arithmetic intensity of algorithms for computing finite element operators increases with increasing polynomial degree. This has made high degree methods particularly attractive on modern CPU and GPU architectures, since on these architectures performance at low degree is limited (severely) by the available memory bandwidth and only a very small fraction of the floating point capacity of the processor is used. Higher degree methods can exploit a significantly greater fraction of the available compute power of modern architectures. However, whilst stable methods for computing high-degree finite element bases are well-established, there is no universal and automated algorithm for the efficient construction of the degree-of-freedom map for arbitrary degree elements. We address this with a new algorithm that can be used in computing degree-of-freedom maps for an arbitrary Ciarlet-type finite element using only the element’s definition and properties of the reference cell, and without requiring a specific implementation for each element. This method is implemented in the library Basix, a component of the FEniCSx libraries. As well as allowing vast simplifications of parts of a codebase, the algorithm allows for new elements to be implemented with ease and has allowed us to support user-defined custom elements that a user can create at runtime without requiring the user to input any information about transformations required to construct a degree-of-freedom map.  \nCCS Concepts: • Mathematics of computing → Mathematical software; Computations on matrices; • Computing methodologies → Linear algebra algorithms.  \nAdditional Key Words and Phrases: finite element methods, degree-of-freedom transformations  \nACM Reference Format:  \nMatthew W. Scroggs and Garth N. Wells. 2026. Algorithm XXXX: Computation of finite element degree-of-freedom transformation  \nmatrices. 1, 1 (July 2026), 22 pages. [https://doi.org/XXXXXXX.XXXXXXX](https://doi.org/XXXXXXX.XXXXXXX)  \n1 INTRODUCTION  \nIn finite element libraries it is usual for global finite element vectors or matrices to be computed by evaluating cell-wise contributions and combining these to form a global vector or matrix. The scattering of cell-wise contributions to the global vector/matrix must preserve the required continuity of finite element functions between cells. The local-to-global map that ensures this continuity is often referred to as the degree-of-freedom map.  \nDegrees-of-freedom (DOFs) of an element can be associated with cell (sub-)entities, i.e. vertices, edges, faces or the cell volume. When using higher-degree finite element spaces, there can be multiple DOFs associated with sub-entities that are shared by more than one cell (e.g., in a degree 3 Lagrange space on a triangle or quadrilateral, there are two DOFs associated with each edge, and edges can be shared by two cells). To ensure the required continuity between cells,  \nAuthors’ Contact Information: Matthew W. Scroggs, [matthew.scroggs.14@ucl.ac.uk](matthew.scroggs.14@ucl.ac.uk), Advanced Research Computing Centre, University College London, London, United Kingdom; Garth N. Wells, [gnw20@cam.ac.uk](gnw20@cam.ac.uk), Department of Engineering, University of Cambridge, Cambridge, United Kingdom.  \nPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy ","cbCaiskwZZA87fuZ","https://ap.wps.com/l/cbCaiskwZZA87fuZ","pdf",742561,1,22,"English","en",105,"# Introduction\n## Degree-of-freedom maps and continuity\n## DOF placement on shared sub-entities\n## Orientation agreement and consequences\n## Prior permutation/transform approaches","[{\"question\":\"Why are high-degree finite element methods especially attractive on modern processors?\",\"answer\":\"Their higher polynomial degree can exploit a larger fraction of available compute power, whereas low-degree performance is often limited by memory bandwidth and underuses floating point capacity.\"},{\"question\":\"What problem does the document target regarding degree-of-freedom (DOF) maps?\",\"answer\":\"It addresses the absence of a universal automated algorithm for efficiently constructing the DOF map for arbitrary degree elements.\"},{\"question\":\"How does the proposed algorithm construct DOF transformation information?\",\"answer\":\"It computes DOF maps for any Ciarlet-type finite element using only the element’s definition and reference-cell properties, without needing a dedicated implementation per element, and it is implemented in Basix within FEniCSx.\"}]",1784186794,55,{"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},"algorithm-xxxx-computation-of-finite-element-degree-of-freedom-transformation-matrices","",{"@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/algorithm-xxxx-computation-of-finite-element-degree-of-freedom-transformation-matrices/83333/",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},"Why are high-degree finite element methods especially attractive on modern processors?","Question",{"text":75,"@type":76},"Their higher polynomial degree can exploit a larger fraction of available compute power, whereas low-degree performance is often limited by memory bandwidth and underuses floating point capacity.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What problem does the document target regarding degree-of-freedom (DOF) maps?",{"text":80,"@type":76},"It addresses the absence of a universal automated algorithm for efficiently constructing the DOF map for arbitrary degree elements.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed algorithm construct DOF transformation information?",{"text":84,"@type":76},"It computes DOF maps for any Ciarlet-type finite element using only the element’s definition and reference-cell properties, without needing a dedicated implementation per element, and it is implemented in Basix within 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