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Domain decomposition solvers for nonlinear multiharmonic finite element equations 非线性多谐有限元方程的域分解求解器
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/jnum.2010.008
D. Copeland, U. Langer
{"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}
引用次数: 30
Elastoviscoplastic Finite Element analysis in 100 lines of Matlab 弹粘塑性有限元分析中的100行Matlab
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/JNMA.2002.157
C. Carstensen, R. Klose
{"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}
引用次数: 43
A posteriori error majorants for approximations of the evolutionary Stokes problem 对于进化斯托克斯问题的近似，后验误差是主要的
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/jnum.2010.005
P. Neittaanmäki, S. Repin
{"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}
引用次数: 10
Numerical method for elliptic multiscale problems 椭圆型多尺度问题的数值方法
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/JNUM.2008.006
I. Greff, W. Hackbusch
{"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}
引用次数: 2
On the construction of a vertex space preconditioner for Morley element Morley元的顶点空间预条件的构造
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/JNMA.2001.295
Jianguo Huang
{"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}
引用次数: 1
Snapshot location by error equilibration in proper orthogonal decomposition for linear and semilinear parabolic partial differential equations 线性和半线性抛物型偏微分方程适当正交分解中的误差平衡快照定位
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/jnum-2014-0001
R. Hoppe, Z. Liu
{"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}
引用次数: 14
On the efficient convolution with the Newton potential 关于牛顿势的有效卷积
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/jnum.2010.013
W. Hackbusch, Kishore Kumar Naraparaju, J. Schneider
{"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}
引用次数: 2
Preconditioned iterative methods for algebraic saddle-point problems 代数鞍点问题的预条件迭代法
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/JNUM.2009.005
Y. Kuznetsov
{"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}
引用次数: 7
Optimal spatial error estimates for DG time discretizations DG时间离散的最优空间误差估计
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/jnum-2013-0009
M. Vlasák
{"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}
引用次数: 2
Moving meshes with freefem++ 用freefem++移动网格
J. Num. Math. Pub Date : 1900-01-01 DOI: 10.1515/jnum-2012-0010
A. Decoene, B. Maury
{"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}
引用次数: 12
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