Abstract In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure.
{"title":"Domain decomposition solvers for nonlinear multiharmonic finite element equations","authors":"D. Copeland, U. Langer","doi":"10.1515/jnum.2010.008","DOIUrl":"https://doi.org/10.1515/jnum.2010.008","url":null,"abstract":"Abstract In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130565119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper provides a short Matlab implementation with documentation of the P 1 finite element method for the numerical solution of viscoplastic and elastoplastic evolution problems in 2D and 3D for von-Mises yield functions and Prandtl-Reuß flow rules. The material behaviour includes perfect plasticity as well as isotropic and kinematic hardening with or without a viscoplastic penalisation in a dual model, i.e. with displacements and the stresses as the main variables. The numerical realisation, however, eliminates the internal variables and becomes displacement-oriented in the end. Any adaption from the given three time-depending examples to more complex applications can easily be performed because of the shortness of the program and the given documentation. In the numerical 2D and 3D examples an efficient error estimator is realized to monitor the stress error.
{"title":"Elastoviscoplastic Finite Element analysis in 100 lines of Matlab","authors":"C. Carstensen, R. Klose","doi":"10.1515/JNMA.2002.157","DOIUrl":"https://doi.org/10.1515/JNMA.2002.157","url":null,"abstract":"Abstract This paper provides a short Matlab implementation with documentation of the P 1 finite element method for the numerical solution of viscoplastic and elastoplastic evolution problems in 2D and 3D for von-Mises yield functions and Prandtl-Reuß flow rules. The material behaviour includes perfect plasticity as well as isotropic and kinematic hardening with or without a viscoplastic penalisation in a dual model, i.e. with displacements and the stresses as the main variables. The numerical realisation, however, eliminates the internal variables and becomes displacement-oriented in the end. Any adaption from the given three time-depending examples to more complex applications can easily be performed because of the shortness of the program and the given documentation. In the numerical 2D and 3D examples an efficient error estimator is realized to monitor the stress error.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132959456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper is concerned with computable estimates of the difference between exact solutions of initial-boundary value problems generated by the Stokes equation and an arbitrary function from the corresponding energy space. They provide guaranteed upper bounds of errors in terms of weighted norms defined on the space-time cylinder where the exact solution is considered. Estimates are derived with the help of techniques based on a transformation of integral identities that was earlier applied (see [Repin, A posteriori Estimates for Partial Differential Equations, Walter de Gruyter, 2008] and the references therein) to the stationary Stokes problem. In this paper, two types of error majorants are derived. They are explicitly computable and contain only global constants. It is proved that the estimates vanish if and only if the functions considered coincide with exact solutions.
摘要本文研究了由Stokes方程生成的初边值问题精确解与任意函数在相应能量空间中的差值的可计算估计。它们根据在考虑精确解的时空柱面上定义的加权规范提供了保证的误差上界。估计是在先前应用的基于积分恒等式变换的技术的帮助下导出的(参见[Repin，偏微分方程的后检估计，Walter de Gruyter, 2008]以及其中的参考文献)，用于平稳Stokes问题。本文导出了两种类型的主误差。它们是显式可计算的，并且只包含全局常量。证明了当且仅当所考虑的函数与精确解重合时估计消失。
{"title":"A posteriori error majorants for approximations of the evolutionary Stokes problem","authors":"P. Neittaanmäki, S. Repin","doi":"10.1515/jnum.2010.005","DOIUrl":"https://doi.org/10.1515/jnum.2010.005","url":null,"abstract":"Abstract This paper is concerned with computable estimates of the difference between exact solutions of initial-boundary value problems generated by the Stokes equation and an arbitrary function from the corresponding energy space. They provide guaranteed upper bounds of errors in terms of weighted norms defined on the space-time cylinder where the exact solution is considered. Estimates are derived with the help of techniques based on a transformation of integral identities that was earlier applied (see [Repin, A posteriori Estimates for Partial Differential Equations, Walter de Gruyter, 2008] and the references therein) to the stationary Stokes problem. In this paper, two types of error majorants are derived. They are explicitly computable and contain only global constants. It is proved that the estimates vanish if and only if the functions considered coincide with exact solutions.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"518 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115862936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we are interested in the coarse-mesh approximations of a class of second order elliptic operators with rough or rapidly oscillatory coefficients. We intend to provide a smoother elliptic operator which on a coarse mesh behaves like the original operator. Note that there is no requirement on smoothness or periodicity of the coefficients. To simplify the theory and the numerical implementations, we restrict ourselves to the one-dimensional case.
{"title":"Numerical method for elliptic multiscale problems","authors":"I. Greff, W. Hackbusch","doi":"10.1515/JNUM.2008.006","DOIUrl":"https://doi.org/10.1515/JNUM.2008.006","url":null,"abstract":"Abstract In this paper we are interested in the coarse-mesh approximations of a class of second order elliptic operators with rough or rapidly oscillatory coefficients. We intend to provide a smoother elliptic operator which on a coarse mesh behaves like the original operator. Note that there is no requirement on smoothness or periodicity of the coefficients. To simplify the theory and the numerical implementations, we restrict ourselves to the one-dimensional case.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115862969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, based on a specially chosen domain decomposition, we construct an overlapping additive Schwarz preconditioner according to the framework of [Brenner, Numer. Math. 72: 419–447, 1996] for the Morley element and show that its condition number is optimal; we analyze in details the structure of this preconditioner, and after proper choices of inexact solvers, we obtain a vertex space preconditioner for the Morley element. Compared with the preconditioners constructed in [Huang, J. Comp. Math. 17: 615–628, 1999, Shi and Xie, J. Comp. Math. 16: 289–304, 1998, Xie, Domain Decomposition and Multigrid Methods for Nonconforming Plate Elements, Chinese Academy of Sciences, 1998], this preconditioner has some advantages, i.e., the computational cost adds little, but the condition number improves greatly.
{"title":"On the construction of a vertex space preconditioner for Morley element","authors":"Jianguo Huang","doi":"10.1515/JNMA.2001.295","DOIUrl":"https://doi.org/10.1515/JNMA.2001.295","url":null,"abstract":"Abstract In this paper, based on a specially chosen domain decomposition, we construct an overlapping additive Schwarz preconditioner according to the framework of [Brenner, Numer. Math. 72: 419–447, 1996] for the Morley element and show that its condition number is optimal; we analyze in details the structure of this preconditioner, and after proper choices of inexact solvers, we obtain a vertex space preconditioner for the Morley element. Compared with the preconditioners constructed in [Huang, J. Comp. Math. 17: 615–628, 1999, Shi and Xie, J. Comp. Math. 16: 289–304, 1998, Xie, Domain Decomposition and Multigrid Methods for Nonconforming Plate Elements, Chinese Academy of Sciences, 1998], this preconditioner has some advantages, i.e., the computational cost adds little, but the condition number improves greatly.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124987996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is well-known that the performance of snapshot based POD and POD-DEIM for spatially semidiscretized parabolic PDEs depends on the proper selection of the snapshot locations. In this contribution, we present an approach that for a fixed number of snapshots selects the location based on error equilibration in the sense that the global discretization error is approximately the same in each associated subinterval. The global discretization error is assessed by a hierarchical-type a posteriori error estimator known from automatic time-stepping for systems of ODEs. We study the impact of this snapshot selection on error equilibration for the ROM and provide numerical examples that illustrate the performance of the suggested approach.
{"title":"Snapshot location by error equilibration in proper orthogonal decomposition for linear and semilinear parabolic partial differential equations","authors":"R. Hoppe, Z. Liu","doi":"10.1515/jnum-2014-0001","DOIUrl":"https://doi.org/10.1515/jnum-2014-0001","url":null,"abstract":"It is well-known that the performance of snapshot based POD and POD-DEIM for spatially semidiscretized parabolic PDEs depends on the proper selection of the snapshot locations. In this contribution, we present an approach that for a fixed number of snapshots selects the location based on error equilibration in the sense that the global discretization error is approximately the same in each associated subinterval. The global discretization error is assessed by a hierarchical-type a posteriori error estimator known from automatic time-stepping for systems of ODEs. We study the impact of this snapshot selection on error equilibration for the ROM and provide numerical examples that illustrate the performance of the suggested approach.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"2015 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127751278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
W. Hackbusch, Kishore Kumar Naraparaju, J. Schneider
Abstract The convolution , where ƒ is smooth, except for some local singularities, arises for example in electronic structure calculations. An efficient convolution with the Newton potential in d dimensions has been proposed in [Hackbusch, Numer. Math. 110: 449–489, 2008]. The convolution is approximated on a refined grid and additional approximations are introduced for efficient evaluation. This paper studies the performance of the method and a precise error analysis of the method is discussed.
{"title":"On the efficient convolution with the Newton potential","authors":"W. Hackbusch, Kishore Kumar Naraparaju, J. Schneider","doi":"10.1515/jnum.2010.013","DOIUrl":"https://doi.org/10.1515/jnum.2010.013","url":null,"abstract":"Abstract The convolution , where ƒ is smooth, except for some local singularities, arises for example in electronic structure calculations. An efficient convolution with the Newton potential in d dimensions has been proposed in [Hackbusch, Numer. Math. 110: 449–489, 2008]. The convolution is approximated on a refined grid and additional approximations are introduced for efficient evaluation. This paper studies the performance of the method and a precise error analysis of the method is discussed.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115238628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we consider the preconditioned Lanczos method for the numerical solution of algebraic systems with singular saddle point matrices. These systems arise from algebraic systems with singularly perturbed symmetric positive definite matrices. The original systems are replaced by equivalent systems with saddle point matrices. Two approaches are proposed to design preconditioners for singular saddle point matrices. The algorithms are applied to the diffusion equation with strongly heterogeneous and anisotropic diffusion tensors.
{"title":"Preconditioned iterative methods for algebraic saddle-point problems","authors":"Y. Kuznetsov","doi":"10.1515/JNUM.2009.005","DOIUrl":"https://doi.org/10.1515/JNUM.2009.005","url":null,"abstract":"Abstract In this paper, we consider the preconditioned Lanczos method for the numerical solution of algebraic systems with singular saddle point matrices. These systems arise from algebraic systems with singularly perturbed symmetric positive definite matrices. The original systems are replaced by equivalent systems with saddle point matrices. Two approaches are proposed to design preconditioners for singular saddle point matrices. The algorithms are applied to the diffusion equation with strongly heterogeneous and anisotropic diffusion tensors.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115826342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper a general parabolic problem is considered and discretized by the discontinuous Galerkin (DG) method in time and, in general, in space. Optimal a priori error estimates in space as well as in time are derived and applied to the heat equation and to a nonlinear convection-diffusion equation.
{"title":"Optimal spatial error estimates for DG time discretizations","authors":"M. Vlasák","doi":"10.1515/jnum-2013-0009","DOIUrl":"https://doi.org/10.1515/jnum-2013-0009","url":null,"abstract":"Abstract In this paper a general parabolic problem is considered and discretized by the discontinuous Galerkin (DG) method in time and, in general, in space. Optimal a priori error estimates in space as well as in time are derived and applied to the heat equation and to a nonlinear convection-diffusion equation.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"281 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123372780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract - The Arbitrary Lagrangian-Eulerian framework allows to compute free surface flows with the Finite Element functions defined on a fittedmesh which follows the globalmotion of the fluid domain. We describe here how freefem++ can be used to implement this method, and we provide two and three dimensional illustrations in the context of water waves.
{"title":"Moving meshes with freefem++","authors":"A. Decoene, B. Maury","doi":"10.1515/jnum-2012-0010","DOIUrl":"https://doi.org/10.1515/jnum-2012-0010","url":null,"abstract":"Abstract - The Arbitrary Lagrangian-Eulerian framework allows to compute free surface flows with the Finite Element functions defined on a fittedmesh which follows the globalmotion of the fluid domain. We describe here how freefem++ can be used to implement this method, and we provide two and three dimensional illustrations in the context of water waves.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124879222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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