[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83649-en":3,"doc-seo-83649-105":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},83649,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Block Preconditioning for Shifted Boundary Method Discretisations of the Stokes Problem","The Shifted Boundary Method (SBM) enforces boundary conditions on a surrogate boundary while compensating the displacement via Taylor expansions, enabling unfitted discretisations for incompressible Stokes flow. Scalable iterative solvers for the resulting non-symmetric systems remain limited. A block preconditioner is proposed that couples the velocity block with a pressure mass-matrix Schur complement approximation. Field-of-values analysis shows SBM terms act as asymptotically small perturbations, giving mesh-independent GMRES on sufficiently fine meshes, with a coarse-grid regime increasing iterations until geometry resolution is adequate.","arXiv :2607 .02336v1 [math .NA] 2 Jul 2026  \nBlock Preconditioning for Shifted Boundary Method Discretisations of the  \nStokes Problem  \nMichał Wichrowskia,∗, Ajay Ajitha  \na Interdisciplinary Center for Scientific Computing, Heidelberg University, Germany  \nAbstract  \nThe Shifted Boundary Method (SBM) sidesteps body-fitted meshing by shifting boundary conditions onto a surrogate boundary and correcting for the displacement through Taylor expansions. Despite its broad analysis and application, scalable iterative solvers for the incompressible Stokes equations remain underdeveloped. We present a block preconditioner for SBM–Stokes discretisations that uses the velocity block together with a pressure mass matrix as a Schur complement approximation. Because the SBM system is non-symmetric, classical operator preconditioning does not apply directly; a field-of-values analysis instead shows that the non-symmetric SBM contributions act as asymptotically small perturbations of a standard saddle-point operator, yielding mesh-independent GMRES convergence on sufficiently fine meshes. Numerical experiments demonstrate iteration counts under refinement across geometries of increasing complexity. We expose a coarse-mesh regime in which an under-resolved grid produces elevated iteration counts, an artefact of insufficient resolution that vanishes once the mesh captures the geometry.  \nKeywords: immersed boundary methods, finite element methods, Shifted Boundary Method, Stokes equations, block preconditioning, saddle-point systems  \n1. Introduction  \nCapturing the intricate detail of real-world geometries within a computational grid is a persistent challenge in the simulation of incompressible flow. The traditional finite element route demands a body-fitted mesh whose cells conform to the domain boundary. While effective, this bespoke mesh generation is computationally expensive and frequently becomes the bottleneck of the entire pipeline for intricate three-dimensional shapes [32], requiring manual intervention to keep elements near curved or narrow boundaries from degenerating into ill-shaped slivers. Unfitted finite element methods offer an alternative by employing a fixed background mesh that remains independent of the physical boundary, trading bespoke meshing for a more complex algebraic structure.  \nAmong these unfitted approaches, the Shifted Boundary Method (SBM) [32] shifts the location where boundary conditions are applied to meet the mesh. By defining the problem on a surrogate domain and extrapolating boundary conditions onto it (typically via Taylor expansions or more general extension operators [50], enforced in a Nitsche-like manner [36]), SBM avoids the complex geometric intersections of methods such as CutFEM [15] and the associated cut-cell quadrature. However, SBM shifts part of the computational challenge from the mesh generator to the linear solver: the extrapolation terms introduce non-symmetry and potential indefiniteness, so a simple mesh no longer guarantees a simple matrix. While the conditioning of SBM scales like O (h−2) [7], comparably to body-fitted methods [16], the efficient solution of the algebraic systems arising from high-order SBM formulations remains largely unexplored.  \nThe SBM has matured rapidly since its first formulation [32, 33] . The original construction used cells strictly within the domain, whereas more recent approaches [48] also admit intersected cells based on a volume-fraction threshold. A complete stability and convergence theory now covers the Stokes problem [9], the Poisson problem on domains with corners [6], and high-order discretisations of arbitrary polynomial degree [10] . The method has been extended to Isogeometric Analysis [1], to a broad range of physics  \n∗ Corresponding [author.](author. mwichro@mimuw.edu.pl)[ mwichro@mimuw.edu.pl](author. mwichro@mimuw.edu.pl)  \nincluding solid mechanics [7, 8], and to problems with embedded interfaces [29, 46], where the formulation is mo","cbCaiktE1bqUZNLi","https://ap.wps.com/l/cbCaiktE1bqUZNLi","pdf",1565564,4,1,21,"English","en",105,"# Introduction\n## Unfitted finite element methods and SBM\n## Non-symmetry, conditioning, and solver challenges\n## Multigrid and AMG background","[{\"question\":\"What problem does the Shifted Boundary Method (SBM) address in Stokes discretisations?\",\"answer\":\"SBM avoids body-fitted meshing by shifting boundary conditions onto a surrogate boundary and correcting the displacement through Taylor expansions, producing unfitted discretisations for incompressible flow.\"},{\"question\":\"Why do classical operator preconditioners not directly apply to SBM–Stokes systems?\",\"answer\":\"The SBM system is non-symmetric, and its extrapolation contributions can make the operator potentially indefinite, so standard symmetric operator preconditioning does not carry over directly.\"},{\"question\":\"How does the proposed block preconditioner approximate the SBM–Stokes Schur complement?\",\"answer\":\"It uses the velocity block together with a pressure mass matrix as a Schur complement approximation, enabling effective iterative solution via GMRES.\"}]",1784189503,53,{"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},"block-preconditioning-for-shifted-boundary-method-discretisations-of-the-stokes-problem","",{"@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/block-preconditioning-for-shifted-boundary-method-discretisations-of-the-stokes-problem/83649/",{"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","2026-07-27","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 does the Shifted Boundary Method (SBM) address in Stokes discretisations?","Question",{"text":75,"@type":76},"SBM avoids body-fitted meshing by shifting boundary conditions onto a surrogate boundary and correcting the displacement through Taylor expansions, producing unfitted discretisations for incompressible flow.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why do classical operator preconditioners not directly apply to SBM–Stokes systems?",{"text":80,"@type":76},"The SBM system is non-symmetric, and its extrapolation contributions can make the operator potentially indefinite, so standard symmetric operator preconditioning does not carry over directly.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed block preconditioner approximate the SBM–Stokes Schur complement?",{"text":84,"@type":76},"It uses the velocity block together with a pressure mass matrix as a Schur complement approximation, enabling effective iterative solution via 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