[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83814-en":3,"doc-seo-83814-105":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},83814,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Mesh Uniform Stability and Error Estimates for HDG Discretizations of the Helmholtz Equation","Revisits the hybridizable discontinuous Galerkin (HDG) method for the Helmholtz equation with a first-order absorbing boundary condition. Prior analysis proves stability without a direct coupling constraint between mesh size and wave number, yet the explicit bound contains negative mesh powers, leading to mesh-dependent blow-up. This work shows the blow-up is a proof artifact and proves a generalized stability estimate with a constant independent of mesh size and uniformly local in κ. The approach uses compactness via HDG discrete distributional gradients and Crouzeix–Raviart lifting, yielding well-posedness, L2 minimal-data convergence, and projection-based error estimates without negative mesh powers, confirmed by numerics.","arXiv :2607 .04507v1 [math .NA] 5 Jul 2026  \nMesh-Uniform Stability and Error Estimates for HDG Discretizations of the Helmholtz Equation  \nYukun Yue∗  \nAbstract  \nWe revisit the hybridizable discontinuous Galerkin method of Cui and Zhang for the Helmholtz equation with first-order absorbing boundary condition. Their analysis proves stability without imposing a mesh constraint coupling h and κ, but the explicit stability bound still contains negative powers of the mesh size. We prove that this mesh-dependent blow-up is not intrinsic to the HDG discretization. Under the same geometric and mesh framework as in the reference analysis, for fixed polynomial degree and for stabilization parameters uniformly bounded above and below, we establish a generalized stability estimate whose constant is independent of themesh size. For every prescribed compact interval 0 \u003C κ0 ≤ κ ≤ κ 1 \u003C ∞ , the stability constant can be chosen locally uniformly with respect to κ . The proof replaces the Rellich-identity and inverse-estimate argument by a compactness argument based on the HDG discrete distributional gradient and the Crouzeix–Raviart lifting mechanism. As consequences, we obtain well-posedness, convergence for minimal L2 data, and projection-based error estimates in which the negative powers of the mesh size appearing in the previous theory are removed. Numerical experiments confirm the predicted mesh-uniform behavior and show a clear contrast with the mesh-dependent growth suggested by the previous explicit bound.  \n1 Introduction  \nThe Helmholtz equation is a prototypical noncoercive time-harmonic wave problem: the associated variational form contains the negative mass term −κ2 (u, v) and is not coercive on H 1 (Ω) in the standard elliptic variational sense; see, for example,[13, Chapter 6] . Its finite element approximation is delicate because the stability of the continuous resolvent, the pollution effect, and the preasymptotic behavior of the discrete scheme all depend on the wave number; see, for example,[2, 14 , 19] . The regularity of the solution, including the high-frequency behavior of corner singularities on nonsmooth domains, further affects the attainable convergence rates [5] . The Helmholtz equation has also motivated a broad computational and inverse-problem literature, including absorbing-boundary and iterative solvers [3], fast multipole acceleration [18], and high-frequency inverse scattering asymptotics [8] .  \nHybridizable discontinuous Galerkin (HDG) methods are attractive in this setting because they retain a local discontinuous structure while reducing the global unknowns to skeletal traces. General HDG frameworks, projection tools, and a posteriori error estimation are developed in [1, 9 , 10 , 12] . Beyond wave problems, HDG methods have been analyzed for convection-dominated diffusion [16], for Stokes and continuum mechanics problems including linear elasticity [15, 24 , 25], for incompressible Navier–Stokes and pressure-robust formulations [4, 21], for Cahn–Hilliard equations [7], and for miscible displacement under minimal regularity [22] . For Helmholtz problems, a range of  \n∗ Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA. Email: [yyue24@wisc.edu](yyue24@wisc.edu).  \nhybrid and hybridizable formulations has been studied, including a multiscale hybrid-mixed method for heterogeneous media [6] and an HDG method with characteristic variables [23] . In particular,[17] analyzed an HDG method for the interior Dirichlet problem, and Cui and Zhang [11] analyzed an HDG method for the Helmholtz equation with first-order absorbing boundary condition in two and three dimensions. The condition hκ2 sufficiently small is mentioned here because it is themesh restriction under which the earlier Dirichlet HDG analysis of [17] obtains its stability and optimal-convergence result; it is stronger than the basic resolution condition hκ ≲ 1. Cui and Zhang’s absorbing-boundary HDG result is therefor","cbCaivIHPJ74M6oy","https://ap.wps.com/l/cbCaivIHPJ74M6oy","pdf",700240,4,1,28,"English","en",105,"# Introduction\n## Background on the Helmholtz problem\n## HDG methods for Helmholtz and related analyses\n## Why the previous explicit stability bound deteriorates\n## Main objective and proof strategy","[{\"question\":\"What problem and numerical method does the paper focus on?\",\"answer\":\"The paper studies the Helmholtz equation using the hybridizable discontinuous Galerkin (HDG) method with a first-order absorbing boundary condition.\"},{\"question\":\"Why does earlier theory predict a mesh-dependent blow-up?\",\"answer\":\"The explicit stability bound from the reference analysis contains negative powers of the mesh size, and the same factor propagates into the projection and L2 error estimates.\"},{\"question\":\"How does the paper remove the negative mesh powers from the stability and error bounds?\",\"answer\":\"It replaces the Rellich-identity plus inverse-estimate mechanism with a compactness argument using the HDG discrete distributional gradient and the Crouzeix–Raviart lifting, leading to mesh-independent stability constants (locally uniform in κ) and error estimates without negative mesh powers.\"}]",1784190589,71,{"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},"mesh-uniform-stability-and-error-estimates-for-hdg-discretizations-of-the-helmholtz-equation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/mesh-uniform-stability-and-error-estimates-for-hdg-discretizations-of-the-helmholtz-equation/83814/",{"url":52,"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","2026-07-24","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 problem and numerical method does the paper focus on?","Question",{"text":75,"@type":76},"The paper studies the Helmholtz equation using the hybridizable discontinuous Galerkin (HDG) method with a first-order absorbing boundary condition.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does earlier theory predict a mesh-dependent blow-up?",{"text":80,"@type":76},"The explicit stability bound from the reference analysis contains negative powers of the mesh size, and the same factor propagates into the projection and L2 error estimates.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper remove the negative mesh powers from the stability and error bounds?",{"text":84,"@type":76},"It replaces the Rellich-identity plus inverse-estimate mechanism with a compactness argument using the HDG discrete distributional gradient and the Crouzeix–Raviart lifting, leading to mesh-independent stability constants (locally uniform in κ) and error estimates without negative mesh powers.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"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":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"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"]