Bosco García-Archilla, Volker John, Sarah Katz, Julia Novo
Abstract Reduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods.
{"title":"POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solution: Error bounds for the pressure","authors":"Bosco García-Archilla, Volker John, Sarah Katz, Julia Novo","doi":"10.1515/jnma-2023-0039","DOIUrl":"https://doi.org/10.1515/jnma-2023-0039","url":null,"abstract":"Abstract Reduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider the Fokker–Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space. The NM time-dependent unknowns of our finite element approximations are probabilities corresponding to discrete space locations and orientation angles. The framework of alternating-direction methods enables us to update the numerical solution in parallel by solving N evolution equations on the sphere and M three-dimensional advection equations in each (pseudo-)time step. To ensure positivity preservation as well as the normalization property of the probability density, we perform algebraic flux correction for each equation and synchronize the correction factors corresponding to different orientation angles. The velocity field for the spatial advection step is obtained using a Schur complement method to solve a generalized system of the incompressible Navier–Stokes equations (NSE). Fiber-induced subgrid-scale effects are taken into account using an effective stress tensor that depends on the second- and fourth-order moments of the orientation density function. Numerical studies are performed for individual subproblems and for the coupled FPE-NSE system.
{"title":"Efficient numerical solution of the Fokker-Planck equation using physics-conforming finite element methods","authors":"Katharina Wegener, D. Kuzmin, S. Turek","doi":"10.1515/jnma-2023-0017","DOIUrl":"https://doi.org/10.1515/jnma-2023-0017","url":null,"abstract":"Abstract We consider the Fokker–Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space. The NM time-dependent unknowns of our finite element approximations are probabilities corresponding to discrete space locations and orientation angles. The framework of alternating-direction methods enables us to update the numerical solution in parallel by solving N evolution equations on the sphere and M three-dimensional advection equations in each (pseudo-)time step. To ensure positivity preservation as well as the normalization property of the probability density, we perform algebraic flux correction for each equation and synchronize the correction factors corresponding to different orientation angles. The velocity field for the spatial advection step is obtained using a Schur complement method to solve a generalized system of the incompressible Navier–Stokes equations (NSE). Fiber-induced subgrid-scale effects are taken into account using an effective stress tensor that depends on the second- and fourth-order moments of the orientation density function. Numerical studies are performed for individual subproblems and for the coupled FPE-NSE system.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83622041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Here, we extend the finite distortion problem from bounded domains in ℝ2 to closed genus-zero surfaces in ℝ3 by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M and a unit sphere 𝕊2 by minimizing the total area distortion energy on ̅ℂ. After the minimizer of the total area distortion energy is determined, it is combined with an initial conformal map to determine the equiareal map between the extended planes. From the inverse stereographic projection, we derive the equiareal map between M and 𝕊2. The total area distortion energy is rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres and is decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization function for the computation of spherical equiareal parameterization between M and 𝕊2. In addition, under relatively mild conditions, we verify that our proposed algorithm has asymptotic R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate that the assumptions for convergence always hold and indicate the efficiency, reliability, and robustness of the developed modified stretch energy minimization function.
{"title":"Fundamental Theory and R-linear Convergence of Stretch Energy Minimization for Spherical Equiareal Parameterization","authors":"Tsung-Ming Huang, Wei-Hung Liao, Wen-Wei Lin","doi":"10.1515/jnma-2022-0072","DOIUrl":"https://doi.org/10.1515/jnma-2022-0072","url":null,"abstract":"Abstract Here, we extend the finite distortion problem from bounded domains in ℝ2 to closed genus-zero surfaces in ℝ3 by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M and a unit sphere 𝕊2 by minimizing the total area distortion energy on ̅ℂ. After the minimizer of the total area distortion energy is determined, it is combined with an initial conformal map to determine the equiareal map between the extended planes. From the inverse stereographic projection, we derive the equiareal map between M and 𝕊2. The total area distortion energy is rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres and is decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization function for the computation of spherical equiareal parameterization between M and 𝕊2. In addition, under relatively mild conditions, we verify that our proposed algorithm has asymptotic R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate that the assumptions for convergence always hold and indicate the efficiency, reliability, and robustness of the developed modified stretch energy minimization function.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75867682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Arndt, W. Bangerth, Maximilian Bergbauer, Marco Feder, M. Fehling, Johannes Heinz, T. Heister, L. Heltai, M. Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Bruno Turcksin, David R. Wells, S. Zampini
Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.5.
摘要本文概述了有限元库协议的新特点。II，版本9.5。
{"title":"The deal.II Library, Version 9.5","authors":"D. Arndt, W. Bangerth, Maximilian Bergbauer, Marco Feder, M. Fehling, Johannes Heinz, T. Heister, L. Heltai, M. Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Bruno Turcksin, David R. Wells, S. Zampini","doi":"10.1515/jnma-2023-0089","DOIUrl":"https://doi.org/10.1515/jnma-2023-0089","url":null,"abstract":"Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.5.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79845223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper presents a posteriori error estimate for the weak Galerkin (WG) finite element method used to solve H(curl)-elliptic problems. Firstly, we introduce a WG method for solving H(curl)-elliptic problems and a corresponding residual type error estimator without a stabilization term. Secondly, we establish the reliability of the error estimator by demonstrating that the stabilization term is controlled by the error estimator. We also evaluate the efficiency of the error estimator using standard bubble functions. Finally, we present some numerical results to show the performances of the error estimator in both uniform and adaptive meshes.
{"title":"A posteriori error estimate for a WG method of H(curl)-elliptic problems","authors":"J. Peng, Yingying Xie, L. Zhong","doi":"10.1515/jnma-2023-0014","DOIUrl":"https://doi.org/10.1515/jnma-2023-0014","url":null,"abstract":"Abstract This paper presents a posteriori error estimate for the weak Galerkin (WG) finite element method used to solve H(curl)-elliptic problems. Firstly, we introduce a WG method for solving H(curl)-elliptic problems and a corresponding residual type error estimator without a stabilization term. Secondly, we establish the reliability of the error estimator by demonstrating that the stabilization term is controlled by the error estimator. We also evaluate the efficiency of the error estimator using standard bubble functions. Finally, we present some numerical results to show the performances of the error estimator in both uniform and adaptive meshes.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75011531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial differential equations having a ( p , δ )-structure for some p ∈ (1, ∞) and δ ⩾0. We establish a priori error estimates, which are optimal for all p ∈ (1, ∞) and δ ⩾0, medius error estimates, i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.
{"title":"Error analysis for a Crouzeix–Raviart approximation of the <i>p</i>-Dirichlet problem","authors":"Alex Kaltenbach","doi":"10.1515/jnma-2022-0106","DOIUrl":"https://doi.org/10.1515/jnma-2022-0106","url":null,"abstract":"Abstract In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial differential equations having a ( p , δ )-structure for some p ∈ (1, ∞) and δ ⩾0. We establish a priori error estimates, which are optimal for all p ∈ (1, ∞) and δ ⩾0, medius error estimates, i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135771207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We propose high order conforming and nonconforming immersed hybridized difference (IHD) methods in two and three dimensions for elliptic interface problems. Introducing the virtual to real transformation (VRT), we could obtain a systematic and unique way of deriving arbitrary high order methods in principle. The optimal number of collocating points for imposing interface conditions is proved, and a unique way of constructing the VRT is suggested. Numerical experiments are performed in two and three dimensions. Numerical results achieving up to the 6th order convergence in the L2-norm are presented for the two dimensional case, and a three dimensional example with a 4th order convergence is presented.
{"title":"High order immersed hybridized difference methods for elliptic interface problems","authors":"Y. Jeon","doi":"10.1515/jnma-2023-0011","DOIUrl":"https://doi.org/10.1515/jnma-2023-0011","url":null,"abstract":"Abstract We propose high order conforming and nonconforming immersed hybridized difference (IHD) methods in two and three dimensions for elliptic interface problems. Introducing the virtual to real transformation (VRT), we could obtain a systematic and unique way of deriving arbitrary high order methods in principle. The optimal number of collocating points for imposing interface conditions is proved, and a unique way of constructing the VRT is suggested. Numerical experiments are performed in two and three dimensions. Numerical results achieving up to the 6th order convergence in the L2-norm are presented for the two dimensional case, and a three dimensional example with a 4th order convergence is presented.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81688970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
E. Bachini, P. Brandner, T. Jankuhn, M. Nestler, S. Praetorius, A. Reusken, A. Voigt
Abstract We study the diffusion of tangential tensor-valued data on curved surfaces. For this purpose, several finite-element-based numerical methods are collected and used to solve a tangential surface n -tensor heat flow problem. These methods differ with respect to the surface representation used, the geometric information required, and the treatment of the tangentiality condition. We emphasize the importance of geometric properties and their increasing influence as the tensorial degree changes from n = 0 to n ≥ 1. A specific example is presented that illustrates how curvature drastically affects the behavior of the solution.
{"title":"Diffusion of tangential tensor fields: numerical issues and influence of geometric properties","authors":"E. Bachini, P. Brandner, T. Jankuhn, M. Nestler, S. Praetorius, A. Reusken, A. Voigt","doi":"10.1515/jnma-2022-0088","DOIUrl":"https://doi.org/10.1515/jnma-2022-0088","url":null,"abstract":"Abstract We study the diffusion of tangential tensor-valued data on curved surfaces. For this purpose, several finite-element-based numerical methods are collected and used to solve a tangential surface n -tensor heat flow problem. These methods differ with respect to the surface representation used, the geometric information required, and the treatment of the tangentiality condition. We emphasize the importance of geometric properties and their increasing influence as the tensorial degree changes from n = 0 to n ≥ 1. A specific example is presented that illustrates how curvature drastically affects the behavior of the solution.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135063421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.1515/jnma-2023-frontmatter2
{"title":"Frontmatter","authors":"","doi":"10.1515/jnma-2023-frontmatter2","DOIUrl":"https://doi.org/10.1515/jnma-2023-frontmatter2","url":null,"abstract":"","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136027418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-17DOI: 10.48550/arXiv.2304.08313
B. García-Archilla, V. John, Sarah Katz, J. Novo
Abstract Reduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods.
{"title":"POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solution: Error bounds for the pressure","authors":"B. García-Archilla, V. John, Sarah Katz, J. Novo","doi":"10.48550/arXiv.2304.08313","DOIUrl":"https://doi.org/10.48550/arXiv.2304.08313","url":null,"abstract":"Abstract Reduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79725451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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