[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84779-en":3,"doc-seo-84779-105":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},84779,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Quadrature Rules for Mass and Stiffness Matrices of Finite Elements for the Wave Equation on Simplexes","Mass lumping enables explicit time stepping for finite-element discretisations of the wave equation when the resulting quadrature weights remain positive and spatial accuracy is preserved. New quadrature rules are introduced for 2nd- and 3rd-degree finite elements on the 4-simplex. For stiffness-matrix integration, numerical quadrature can be more efficient than exact evaluation by using fewer nodes. Additional rules are derived in two and four dimensions for lower-degree simplex elements, with extra results in three dimensions.","arXiv :2607 .04995v1 [math .NA] 6 Jul 2026  \nQuadrature rules for the mass and stiffness matrices of finite elements for the wave equation on the 2-, 3-and 4-simplex  \nW. A. Mulder  \nDelft University of Technology, Department of Geoscience & Engineering, Faculty of Civil Engineering and Geosciences, Stevinweg 1, Delft, 2600 GA, The Netherlands  \nAbstract  \nMass lumping enables explicit time stepping for finite-element discretisation of the wave equation if the resulting quadrature weights are positive and accuracy is preserved. New rules for elements of degree two and three on the 4-simplex are presented. Numerical quadrature for the stiffness matrix can be more efficient than exact evaluation if it requires fewer nodes. For the latter, new rules in two and four space dimensions for the lower-degree elements on the simplex were found, as well as some additional results for three dimensions.  \nKeywords: quadrature, simplex, node patterns, polynomial, finite elements, wave equation  \n2000 MSC: 65D32, 65N30  \n1. Introduction  \nWhen solving the wave equation, finite elements can be more efficient than finite differences if the mesh scales with the local (shear) velocity and if the element faces follow the discontinuities of the material properties [1, 2, 3, e.g.] . Mass lumping avoids the inversion the large sparse mass matrix and allows for explicit time stepping, resulting in a scheme with a computational structure similar to the finite-difference method. Straightforward lumping for standard polynomial simplicial elements leads to a loss of spatial accuracy, which can be restored by the use of higher-degree polynomials in the interior of the element, as demonstrated for triangles [4, 5] and tetrahedra [1] .  \nPreprint submitted to J. Comput. Appl. Math., December 5, 2025  \nThe construction of mass-lumped elements involves a choice of quadrature nodes and the requirement that quadrature be exact for a certain set of polynomials. The solution of the resulting polynomial system of equations provides the actual node positions and quadrature weights, but its complexity explodes rapidly when the number of nodes and polynomial degree increases.  \nSo far, triangular elements could be constructed for degrees 2 [6, 7], 3 [8, 5], 4 [1], 5 [9], 6 [10], 7 and 8 [11, 12] . The last two papers also contain degree-9 elements, but the one in [12] has degree 10 instead of 9 on the edges whereas the element in [11], when used as an initial guess, does not seem to converge to a solution of the quadrature equations when very high extended precision is used in a Newton-type root-finding method. Nevertheless, the proposed element appears to be sufficient for applications with double-precision computations.  \nTetrahedral elements were initially found for degree 2 [1] and 3 [9] . A sharper accuracy criterion was proposed in [13], leading to new tetrahedral elements of degree 2 and 3, as well as several of degree 4 . In that paper, the degree-4 elements had a subset of the polynomials with degree higher than 4 . Here, elements of that type will not be considered. The same sharper accuracy criterion also led to simpler triangular elements of degree 5 and 6 [14, 15] .  \nExplicit time stepping for the wave equation involves repeated multiplications of the stiffness matrices and the solution vectors. Evaluation on the fly instead of full assembly of these matrices can speed up the computations [16, e.g.] . Numerical instead of exact quadrature for the stiffness matrices can further reduce the required number of operations [17] .  \nThe accuracy of the time stepping scheme is another topic. One option is higher-order time stepping with the Lax–Wendroff approach [18], also known as the Cauchy–Kowalevsky[19] or Dablain [20] or modified-equation method [21] . Another with low cost is dispersion correction of the second-order scheme [22, 23, 24] .  \nThe goal of the current paper is to extend mass lumping to 4D. In addition, dedicated numerical quadrature rules for","cbCaioxNKjVwhV2H","https://ap.wps.com/l/cbCaioxNKjVwhV2H","pdf",155913,2,1,25,"English","en",105,"# Introduction\n## Mass lumping and explicit time stepping\n## Quadrature-node construction and complexity\n## Relation to existing element rules\n# Quadrature reviewed\n## Mass lumping and diagonal mass matrices\n## Polynomial exactness and symmetric node patterns\n# Results and element rules\n## Lower-degree elements on 2-, 3-, and 4-simplex\n## Stiffness-matrix quadrature in multiple dimensions\n# Conclusion","[{\"question\":\"How does mass lumping support explicit time stepping for the wave equation?\",\"answer\":\"Mass lumping replaces the mass matrix by a diagonal matrix using quadrature weights, avoiding inversion of the large sparse mass matrix. Positivity of these weights enables stable explicit time stepping while preserving spatial accuracy.\"},{\"question\":\"Why can numerical quadrature for the stiffness matrix be preferable to exact evaluation?\",\"answer\":\"Numerical quadrature can be more efficient if it uses fewer nodes than exact evaluation. The paper constructs dedicated quadrature rules to reduce computational effort while maintaining required accuracy criteria.\"},{\"question\":\"What new results are provided for elements on the simplex?\",\"answer\":\"The work presents new quadrature rules for degree-two and degree-three elements on the 4-simplex. It also provides rules for stiffness matrices in two to four dimensions for lower-degree elements, plus additional results for three dimensions.\"}]",1784198182,63,{"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},"quadrature-rules-for-mass-and-stiffness-matrices-of-finite-elements-for-the-wave-equation-on-simplexes","",{"@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/quadrature-rules-for-mass-and-stiffness-matrices-of-finite-elements-for-the-wave-equation-on-simplexes/84779/",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","2026-07-23","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},"How does mass lumping support explicit time stepping for the wave equation?","Question",{"text":75,"@type":76},"Mass lumping replaces the mass matrix by a diagonal matrix using quadrature weights, avoiding inversion of the large sparse mass matrix. Positivity of these weights enables stable explicit time stepping while preserving spatial accuracy.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why can numerical quadrature for the stiffness matrix be preferable to exact evaluation?",{"text":80,"@type":76},"Numerical quadrature can be more efficient if it uses fewer nodes than exact evaluation. The paper constructs dedicated quadrature rules to reduce computational effort while maintaining required accuracy criteria.",{"name":82,"@type":73,"acceptedAnswer":83},"What new results are provided for elements on the simplex?",{"text":84,"@type":76},"The work presents new quadrature rules for degree-two and degree-three elements on the 4-simplex. It also provides rules for stiffness matrices in two to four dimensions for lower-degree elements, plus additional results for three dimensions.","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 & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"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"]