[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86050-en":3,"doc-seo-86050-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},86050,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Solving the Stokes Equations via a Least Squares Weak Galerkin Method","Develops a least-squares weak Galerkin (LS-WG) finite element method for the Stokes equations on general polygonal and polyhedral meshes. The approach uses discrete weak derivatives on discontinuous polynomial spaces to handle complex geometries naturally. The least-squares reformulation eliminates the traditional inf-sup (LBB) compatibility requirement, yielding a symmetric positive definite discrete linear system. The paper proves numerical well-posedness and derives optimal-order error bounds in a discrete energy norm, with convergence O(h^k) for projection error and O(h^{k−1}) for global approximation using degree k velocity and degree k−1 pressure.","arXiv :2607 . 10831v1 [math .NA] 12 Jul 2026  \nSOLVING THE STOKES EQUATIONS VIA A LEAST SQUARES WEAK GALERKIN METHOD  \nCHUNMEI WANG† AND SHANGYOU ZHANG  \nAbstract. We present a least-squares weak Galerkin (LS-WG) finite element method for solving the Stokes equations on arbitrary polygonal and polyhedral meshes. By utilizing discrete weak derivatives on discontinuous polynomial spaces, the proposed framework naturally accommodates complex domain geometries and general partitions. Crucially, this least-squares formulation bypasses the traditional inf-sup (LBB) compatibility condition, transforming the standard indefinite saddle-point problem into an inherently symmetric and positive definite (SPD) discrete linear system. We establish the well-posedness of the numerical scheme and rigorously derive optimal-order error estimates in a custom discrete energy norm. Specifically, we prove convergence rates of O(hk ) for the discrete projection error and O(hk−1) for the global approximation error when employing polynomials of degree k ≥ 1 for the velocity field and k − 1 for the pressure. Extensive numerical experiments confirm these theoretical convergence rates, demonstrating the method’s robustness, geometric flexibility, and overall efficiency.  \n1. Introduction  \nThe Stokes equations are fundamental to fluid mechanics, governing the behavior of incompressible viscous fluids in the creeping flow regime where the Reynolds number is very low. In this regime, viscous forces dominate inertial effects, making the model essential for simulating diverse phenomena across science and engineering. Key applications range from microfluidicsand lab-on-a-chip technologies to biological systems—such as the swimming mechanisms of microorganisms—and large-scale geophysical processes, including mantle convection and glacier dynamics. Furthermore, the Stokes problem provides the underlying linear structure for the full, nonlinear  \nDepartment of Mathematics, University of Florida, Gainesville, FL 32611, USA.  \nDepartment of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA  \nE-mail addresses: [chunmei.wang@ufl.edu](chunmei.wang@ufl.edu) , [szhang@udel.edu](szhang@udel.edu).  \n2020 Mathematics Subject Classification. 65N15, 65N30, 76D07; Secondary, 35B45, 35J50 .  \nKey words and phrases. Weak Galerkin, least squares, finite element methods, the Stokes equations, polyhedral meshes.  \n† Corresponding author.  \n2 LEAST-SQUARES WEAK GALERKIN  \nNavier-Stokes equations. Consequently, the development of robust and geometrically flexible numerical solvers for the Stokes system remains a critical area of computational research.  \nIn this paper, we introduce a least squares weak Galerkin (WG) finite element method for the Stokes equations. We seek a velocity field u and a pressure p that satisfy the following system:  \n(1.1) −∆u + ∇p = f in Ω ,  \n(1.2) ∇ · u = 0 in Ω ,  \n(1.3) u = g on ∂Ω,  \nwhere Ω ⊂ Rd (d = 2 , 3) is a polygonal or polyhedral domain, f is the body force, and g represents the Dirichlet boundary data.  \nThe numerical approximation of the Stokes problem presents significant challenges, primarily due to the indefinite saddle-point nature of the system. Standard mixed finite element methods (FEM) require that the velocity and pressure spaces satisfy the restrictive discrete inf-sup (LadyzhenskayaBabuˇska-Brezzi or LBB) condition to ensure stability and well-posedness [2] . Constructing inf-sup stable element pairs on complex or non-matching grids is often computationally intensive and geometrically limiting.  \nTo overcome these constraints, discontinuous Galerkin (DG) methods were developed to relax inter-element continuity, thereby enhancing geometric flexibility and enabling divergence-free velocity approximations [1] . More recently, the weak Galerkin (WG) finite element method has emerged as a powerful alternative. The WG framework [11, 12, 38, 42, 13, 14, 15, 16, 40, 44, 6, 37, 20, 10, 22, 46, 31, 36, 32, 33, 34, 39, 43, 8","cbCaiq9s0cCI6HYo","https://ap.wps.com/l/cbCaiq9s0cCI6HYo","pdf",336000,1,20,"English","en",105,"# Abstract\n# Introduction\n# Least-Squares Weak Galerkin","[{\"question\":\"What problem does the LS-WG method address?\",\"answer\":\"It targets numerical solution of the Stokes equations for incompressible viscous flow on arbitrary polygonal and polyhedral domains.\"},{\"question\":\"How does the least-squares formulation change the resulting linear system?\",\"answer\":\"It bypasses the inf-sup (LBB) compatibility condition and converts the indefinite saddle-point formulation into a symmetric positive definite (SPD) discrete system.\"},{\"question\":\"What error convergence rates are proved?\",\"answer\":\"Using polynomials of degree k≥1 for velocity and degree k−1 for pressure, the method achieves O(h^k) convergence for discrete projection error and O(h^{k−1}) for global approximation error in a discrete energy norm.\"}]",1784208081,50,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"solving-the-stokes-equations-via-a-least-squares-weak-galerkin-method","",{"@graph":35,"@context":84},[36,53,67],{"@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/solving-the-stokes-equations-via-a-least-squares-weak-galerkin-method/86050/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the LS-WG method address?","Question",{"text":74,"@type":75},"It targets numerical solution of the Stokes equations for incompressible viscous flow on arbitrary polygonal and polyhedral domains.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the least-squares formulation change the resulting linear system?",{"text":79,"@type":75},"It bypasses the inf-sup (LBB) compatibility condition and converts the indefinite saddle-point formulation into a symmetric positive definite (SPD) discrete system.",{"name":81,"@type":72,"acceptedAnswer":82},"What error convergence rates are proved?",{"text":83,"@type":75},"Using polynomials of degree k≥1 for velocity and degree k−1 for pressure, the method achieves O(h^k) convergence for discrete projection error and O(h^{k−1}) for global approximation error in a discrete energy norm.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,113,118,121,125,128,132],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":28,"slug":112},6,"Technology","technology",{"id":114,"doc_module":4,"doc_module_name":45,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":119,"slug":120},30,"research-report",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":21,"slug":124},9,"Religion & Spirituality","religion-spirituality",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":126,"show_sort_weight":21,"slug":127},"World Cup","world-cup",{"id":129,"doc_module":4,"doc_module_name":45,"category_name":130,"show_sort_weight":129,"slug":131},10,"Lifestyle","lifestyle",{"id":133,"doc_module":4,"doc_module_name":45,"category_name":134,"show_sort_weight":105,"slug":135},19,"General","general"]