[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82546-en":3,"doc-seo-82546-105":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},82546,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Goal-Oriented Space-Time Adaptivity for the Navier–Stokes Equations based on the Dual Weighted Residual Method","Goal-oriented a posteriori error estimation combines the Dual Weighted Residual (DWR) method with space-time mesh adaptivity for the Navier–Stokes equations, enabling reliable control of user-defined target quantities on computationally feasible meshes. The resulting nonlinear algebraic systems on space-time slabs are solved via Newton’s method with GMRES, using slab-wise geometric multigrid preconditioning. The approach is implemented in deal.II with MPI parallelization and uses discontinuous Galerkin time discretization and inf-sup stable tensor-product finite element pairs with discontinuous pressure. Benchmark studies assess accuracy, efficiency, and stability.","arXiv :2607 .00686v1 [math .NA] 1 Jul 2026  \nGoal-Oriented Space-Time Adaptivity for the Navier–Stokes Equations based on the Dual Weighted  \nResidual Method  \nMarius Paul Bruchh¨auser∗,‡ · Nils Margenberg† · Markus Bause‡  \nAbstract  \nThis work presents a goal-oriented a posteriori error estimator based on the Dual Weighted Residual (DWR) method together with space-time mesh adaptivity for the Navier–Stokes equations. The resulting nonlinear algebraic systems on the space-time slabs are solved by Newton’s method with GMRES, preconditioned by a slab-wise geometric multigrid method. This combination yields reliable control of target quantities on computationally feasible space-time meshes together with a robust and efficient solution of the algebraic systems. The implementation is based on a MPI-parallel programming model in the deal.II library. Further ingredients are a discontinuous Galerkin discretization in time and inf-sup stable finite element pairs with discontinuous pressure on tensor-product meshes. The performance of the approach is investigated in benchmark computations with regard to accuracy, efficiency, and stability.  \nKeywords: Navier–Stokes Equations · Goal-Oriented Error Control · Dual Weighted Residual Method · Space-Time Adaptivity · Geometric Multigrid Method · Local Vanka Smoother · Nitsche Method  \n1 Introduction  \nThe accurate numerical simulation of incompressible viscous fluid flow governed by the Navier–Stokes equations remains challenging, in particular when strong dynamics and fine-scale structures in space and time have to be resolved. The robust and efficient solution of the arising algebraic systems adds a further layer of complexity. Moreover, algorithms that address these challenges have to be implemented in a software framework that is suitable for computations on modern hardware architectures with tens of thousands of compute cores, including hybrid CPU and GPU systems. This places high demands on the underlying data structures. In this work, we design an approach that addresses these requirements. The implementation is based on the deal.II library [1] .  \nSpace-time mesh adaptation based on rigorous error representations, or on indicators derived from such representations, has been shown to yield accurate results on computationally feasible grids at moderate numerical cost, even for problems with complex solution structures [26, 39] . Among these approaches, the Dual Weighted  \n∗ ,‡ Helmut Schmidt University Hamburg, Faculty of Mechanical and Civil Engineering, Chair of Numerical Mathematics, Holstenhofweg 85, 22043 Hamburg, Germany, [bruchhaeuser@hsu-hamburg.de](bruchhaeuser@hsu-hamburg.de) ( ∗ corresponding author), [bause@hsu-hamburg.de](bause@hsu-hamburg.de)  \n†University of Magdeburg, Institute for Analysis and Numerics, Universit¨atsplatz 2, 39104 Magdeburg, Germany, [nils.margenberg@ovgu.de.de](nils.margenberg@ovgu.de.de)  \nResidual method (DWR) [6] offers the possibility of controlling the discretization error in user-defined goal quantities of practical interest. The key idea of this approach is to compute dual weights for the local residuals entering the error representation. This requires the solution of the generally nonlinear primal problem and its linearized dual problem by space-time finite element methods. The DWR method has been successfully applied to numerous problem classes, including fluid flow [7, 10], wave propagation [3], fluid-structure interaction [33, 43], among others. For a comprehensive review with detailed references on various model problems, we refer, for instance, to [6, 4, 33] . Error control in norms associated with the weak formulation constitutes a conceptually different class of techniques for a posteriori error estimation and automatic mesh adaptation. For this, an approach using equilibrated flux reconstructionsis proposed in [16] .  \nThe DWR concept relies on solving the primal Navier–Stokes system and the associated linearized dual problem by space-","cbCaibUKEbsJ8lOj","https://ap.wps.com/l/cbCaibUKEbsJ8lOj","pdf",1750419,1,28,"English","en",105,"# Introduction\n## Goal-oriented DWR error control and dual weights\n## Discretization choices and stability components\n## Space-time slab localization and solver strategy\n## Scope of benchmark investigations","[{\"question\":\"What is the core method for error estimation in the document?\",\"answer\":\"The document uses a goal-oriented a posteriori error estimator based on the Dual Weighted Residual (DWR) method, where dual weights are computed for local residuals in the error representation.\"},{\"question\":\"How are the nonlinear systems arising from the space-time formulation solved?\",\"answer\":\"Nonlinear algebraic systems on space-time slabs are solved with Newton’s method, where each Newton linearization is handled by GMRES preconditioned by a slab-wise geometric multigrid method.\"},{\"question\":\"What numerical techniques are used for discretization and boundary conditions?\",\"answer\":\"Time is discretized with a discontinuous Galerkin method, space uses inf-sup stable finite element pairs with discontinuous pressure on tensor-product meshes, and nonhomogeneous Dirichlet boundary conditions are imposed via Nitsche’s 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is the core method for error estimation in the document?","Question",{"text":75,"@type":76},"The document uses a goal-oriented a posteriori error estimator based on the Dual Weighted Residual (DWR) method, where dual weights are computed for local residuals in the error representation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the nonlinear systems arising from the space-time formulation solved?",{"text":80,"@type":76},"Nonlinear algebraic systems on space-time slabs are solved with Newton’s method, where each Newton linearization is handled by GMRES preconditioned by a slab-wise geometric multigrid method.",{"name":82,"@type":73,"acceptedAnswer":83},"What numerical techniques are used for discretization and boundary conditions?",{"text":84,"@type":76},"Time is discretized with a discontinuous Galerkin method, space uses inf-sup stable finite element pairs with discontinuous pressure on tensor-product meshes, and nonhomogeneous Dirichlet boundary conditions are 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