Abstract In this paper we propose a time-splitting method for degenerate convection–diffusion equations perturbed stochastically by white noise. This work generalizes previous results on splitting operator techniques for stochastic hyperbolic conservation laws for the degenerate parabolic case. The convergence in Llocp$begin{array}{} displaystyle L^p_{rm loc} end{array}$ of the time-splitting operator scheme to the unique weak entropy solution is proven. Moreover, we analyze the performance of the splitting approximation by computing its convergence rate and showing numerical simulations for some benchmark examples, including a fluid flow application in porous media.
{"title":"Convergence of time-splitting approximations for degenerate convection–diffusion equations with a random source","authors":"Roberto Díaz-Adame, S. Jerez","doi":"10.1515/JNMA-2020-0012","DOIUrl":"https://doi.org/10.1515/JNMA-2020-0012","url":null,"abstract":"Abstract In this paper we propose a time-splitting method for degenerate convection–diffusion equations perturbed stochastically by white noise. This work generalizes previous results on splitting operator techniques for stochastic hyperbolic conservation laws for the degenerate parabolic case. The convergence in Llocp$begin{array}{} displaystyle L^p_{rm loc} end{array}$ of the time-splitting operator scheme to the unique weak entropy solution is proven. Moreover, we analyze the performance of the splitting approximation by computing its convergence rate and showing numerical simulations for some benchmark examples, including a fluid flow application in porous media.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86000817","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 A collocated finite-volume method for the incompressible Navier–Stokes problem is introduced. The method applies to general polyhedral grids and demonstrates higher than the first order of convergence. The velocity components and the pressure are approximated by piecewise-linear continuous and piecewise-constant fields, respectively. The method does not require artificial boundary conditions for pressure but requires stabilization term to suppress the error introduced by piecewise-constant pressure for convection-dominated problems. Both the momentum and continuity equations are approximated in a flux-conservative fashion, i.e., the conservation for both quantities is discretely exact. The attractive side of the method is a simple flux-based finite-volume construction of the scheme. Applicability of the method is demonstrated on several numerical tests using general polyhedral grids.
{"title":"Collocated finite-volume method for the incompressible Navier–Stokes problem","authors":"K. Terekhov","doi":"10.1515/jnma-2020-0008","DOIUrl":"https://doi.org/10.1515/jnma-2020-0008","url":null,"abstract":"Abstract A collocated finite-volume method for the incompressible Navier–Stokes problem is introduced. The method applies to general polyhedral grids and demonstrates higher than the first order of convergence. The velocity components and the pressure are approximated by piecewise-linear continuous and piecewise-constant fields, respectively. The method does not require artificial boundary conditions for pressure but requires stabilization term to suppress the error introduced by piecewise-constant pressure for convection-dominated problems. Both the momentum and continuity equations are approximated in a flux-conservative fashion, i.e., the conservation for both quantities is discretely exact. The attractive side of the method is a simple flux-based finite-volume construction of the scheme. Applicability of the method is demonstrated on several numerical tests using general polyhedral grids.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83078162","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 aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e., for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager–Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one.
{"title":"Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations","authors":"P. Lederer, C. Merdon","doi":"10.1515/jnma-2021-0078","DOIUrl":"https://doi.org/10.1515/jnma-2021-0078","url":null,"abstract":"Abstract This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e., for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager–Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87478349","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, B. Blais, Thomas C. Clevenger, M. Fehling, A. Grayver, T. Heister, L. Heltai, M. Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Reza Rastak, Ignacio Tomas, Bruno Turcksin, Zhuoran Wang, David R. Wells
Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.2.
摘要本文概述了有限元库协议的新特点。II，版本9.2。
{"title":"The deal.II library, Version 9.2","authors":"D. Arndt, W. Bangerth, B. Blais, Thomas C. Clevenger, M. Fehling, A. Grayver, T. Heister, L. Heltai, M. Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Reza Rastak, Ignacio Tomas, Bruno Turcksin, Zhuoran Wang, David R. Wells","doi":"10.1515/jnma-2020-0043","DOIUrl":"https://doi.org/10.1515/jnma-2020-0043","url":null,"abstract":"Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.2.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80015737","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 a BOundary Update using Resolvent (BOUR) partitioned method, second-order accurate in time, unconditionally stable, for the interaction between a viscous incompressible fluid and a thin structure. The method is algorithmically similar to the sequential Backward Euler — Forward Euler implementation of the midpoint quadrature rule. (i) The structure and fluid sub-problems are first solved using a Backward Euler scheme, (ii) the velocities of fluid and structure are updated on the boundary via a second-order consistent resolvent operator, and then (iii) the structure and fluid sub-problems are solved again, using a Forward Euler scheme. The stability analysis based on energy estimates shows that the scheme is unconditionally stable. Error analysis of the semi-discrete problem yields second-order convergence in time. The two numerical examples confirm theoretical convergence analysis results and show an excellent agreement between the proposed partitioned scheme and the monolithic scheme.
{"title":"Boundary update via resolvent for fluid–structure interaction","authors":"M. Bukač, C. Trenchea","doi":"10.1515/jnma-2019-0081","DOIUrl":"https://doi.org/10.1515/jnma-2019-0081","url":null,"abstract":"Abstract We propose a BOundary Update using Resolvent (BOUR) partitioned method, second-order accurate in time, unconditionally stable, for the interaction between a viscous incompressible fluid and a thin structure. The method is algorithmically similar to the sequential Backward Euler — Forward Euler implementation of the midpoint quadrature rule. (i) The structure and fluid sub-problems are first solved using a Backward Euler scheme, (ii) the velocities of fluid and structure are updated on the boundary via a second-order consistent resolvent operator, and then (iii) the structure and fluid sub-problems are solved again, using a Forward Euler scheme. The stability analysis based on energy estimates shows that the scheme is unconditionally stable. Error analysis of the semi-discrete problem yields second-order convergence in time. The two numerical examples confirm theoretical convergence analysis results and show an excellent agreement between the proposed partitioned scheme and the monolithic scheme.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77388445","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 this note we consider an efficient data–sparse approximation of a modified Hilbert type transformation as it is used for the space–time finite element discretization of parabolic evolution equations in the anisotropic Sobolev space H1,1/2(Q). The resulting bilinear form of the first-order time derivative is symmetric and positive definite, and similar as the integration by parts formula for the Laplace hypersingular boundary integral operator in 2D. Hence we can apply hierarchical matrices for data–sparse representations and for acceleration of the computations. Numerical results show the efficiency in the approximation of the first-order time derivative. An efficient realisation of the modified Hilbert transformation is a basic ingredient when considering general space–time finite element methods for parabolic evolution equations, and for the stable coupling of finite and boundary element methods in anisotropic Sobolev trace spaces.
{"title":"A note on the efficient evaluation of a modified Hilbert transformation","authors":"O. Steinbach, Marco Zank","doi":"10.1515/jnma-2019-0099","DOIUrl":"https://doi.org/10.1515/jnma-2019-0099","url":null,"abstract":"Abstract In this note we consider an efficient data–sparse approximation of a modified Hilbert type transformation as it is used for the space–time finite element discretization of parabolic evolution equations in the anisotropic Sobolev space H1,1/2(Q). The resulting bilinear form of the first-order time derivative is symmetric and positive definite, and similar as the integration by parts formula for the Laplace hypersingular boundary integral operator in 2D. Hence we can apply hierarchical matrices for data–sparse representations and for acceleration of the computations. Numerical results show the efficiency in the approximation of the first-order time derivative. An efficient realisation of the modified Hilbert transformation is a basic ingredient when considering general space–time finite element methods for parabolic evolution equations, and for the stable coupling of finite and boundary element methods in anisotropic Sobolev trace spaces.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82706572","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 The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble–Hilbert lemma, we derive a probability law that evaluates the relative accuracy, considered as a random variable, between two finite elements Pk and Pm, k < m. We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes, the second one is dedicated to a sequence of simplexes constituting a given mesh. Both of these applications could be considered as a first step toward application for adaptive mesh refinement with probabilistic methods.
{"title":"On generalized binomial laws to evaluate finite element accuracy: preliminary probabilistic results for adaptive mesh refinement","authors":"J. Chaskalovic, F. Assous","doi":"10.1515/jnma-2019-0001","DOIUrl":"https://doi.org/10.1515/jnma-2019-0001","url":null,"abstract":"Abstract The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble–Hilbert lemma, we derive a probability law that evaluates the relative accuracy, considered as a random variable, between two finite elements Pk and Pm, k < m. We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes, the second one is dedicated to a sequence of simplexes constituting a given mesh. Both of these applications could be considered as a first step toward application for adaptive mesh refinement with probabilistic methods.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85126862","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 extends the applicability of the combined use of the virtual element method (VEM) and the boundary element method (BEM), recently introduced to solve the coupling of linear elliptic equations in divergence form with the Laplace equation, to the case of acoustic scattering problems in 2D and 3D. The well-posedness of the continuous and discrete formulations are established, and then Cea-type estimates and consequent rates of convergence are derived.
{"title":"Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems","authors":"G. Gatica, S. Meddahi","doi":"10.1515/jnma-2019-0068","DOIUrl":"https://doi.org/10.1515/jnma-2019-0068","url":null,"abstract":"Abstract This paper extends the applicability of the combined use of the virtual element method (VEM) and the boundary element method (BEM), recently introduced to solve the coupling of linear elliptic equations in divergence form with the Laplace equation, to the case of acoustic scattering problems in 2D and 3D. The well-posedness of the continuous and discrete formulations are established, and then Cea-type estimates and consequent rates of convergence are derived.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75301930","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 this paper, a two-grid method with backtracking is proposed and investigated for the mixed Stokes/Darcy system which describes a fluid flow coupled with a porous media flow. Based on the classical two-grid method [15], a coarse mesh correction is carried out to derive optimal error bounds for the velocity field and the piezometric head in L2 norm. Finally, results of numerical experiments are provided to support the theoretical results.
{"title":"A two-grid method with backtracking for the mixed Stokes/Darcy model","authors":"Guangzhi Du, Liyun Zuo","doi":"10.1515/jnma-2020-0001","DOIUrl":"https://doi.org/10.1515/jnma-2020-0001","url":null,"abstract":"Abstract In this paper, a two-grid method with backtracking is proposed and investigated for the mixed Stokes/Darcy system which describes a fluid flow coupled with a porous media flow. Based on the classical two-grid method [15], a coarse mesh correction is carried out to derive optimal error bounds for the velocity field and the piezometric head in L2 norm. Finally, results of numerical experiments are provided to support the theoretical results.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86703184","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 develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors’ recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search.
{"title":"Acceleration of nonlinear solvers for natural convection problems","authors":"Sara N. Pollock, L. Rebholz, Mengying Xiao","doi":"10.1515/jnma-2020-0067","DOIUrl":"https://doi.org/10.1515/jnma-2020-0067","url":null,"abstract":"Abstract This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors’ recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77482465","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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