Pub Date : 2005-09-01DOI: 10.1515/156939505774286102
M. Banda, Mohammed Seaïd
We present a higher order generalization for relaxation methods in the framework presented by Jin and Xin in [10]. The schemes employ general higher order integration for spatial discretization and higher order implicit-explicit (IMEX) schemes or Total Variation diminishing (TVD) Runge–Kutta schemes for time integration of relaxing or relaxed schemes, respectively, for time integration. Numerical experiments are performed on various test problems, in particular, the Burger's and Euler equations of inviscid gas dynamics in both one and two space dimensions. In addition, uniform convergence with respect to the relaxation parameter is demonstrated.
{"title":"Higher-order relaxation schemes for hyperbolic systems of conservation laws","authors":"M. Banda, Mohammed Seaïd","doi":"10.1515/156939505774286102","DOIUrl":"https://doi.org/10.1515/156939505774286102","url":null,"abstract":"We present a higher order generalization for relaxation methods in the framework presented by Jin and Xin in [10]. The schemes employ general higher order integration for spatial discretization and higher order implicit-explicit (IMEX) schemes or Total Variation diminishing (TVD) Runge–Kutta schemes for time integration of relaxing or relaxed schemes, respectively, for time integration. Numerical experiments are performed on various test problems, in particular, the Burger's and Euler equations of inviscid gas dynamics in both one and two space dimensions. In addition, uniform convergence with respect to the relaxation parameter is demonstrated.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130330875","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}
Pub Date : 2005-09-01DOI: 10.1515/156939505774286148
C. Bacuta, Jinru Chen, Yunqing Huang, Jinchao Xu, L. Zikatanov
We consider the Stokes problem on a plane polygonal domain We propose a finite element method for overlapping or nonmatching grids for the Stokes problem based on the partition of unity method. We prove that the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains. The results are valid for multiple subdomains and any spatial dimension.
{"title":"Partition of unity method on nonmatching grids for the Stokes problem","authors":"C. Bacuta, Jinru Chen, Yunqing Huang, Jinchao Xu, L. Zikatanov","doi":"10.1515/156939505774286148","DOIUrl":"https://doi.org/10.1515/156939505774286148","url":null,"abstract":"We consider the Stokes problem on a plane polygonal domain We propose a finite element method for overlapping or nonmatching grids for the Stokes problem based on the partition of unity method. We prove that the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains. The results are valid for multiple subdomains and any spatial dimension.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131130175","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}
Pub Date : 2005-09-01DOI: 10.1515/156939505774286120
P. Novati
In this paper we introduce a new class of explicit one-step methods of order 2 that can be used for solving stiff problems. This class constitutes a generalization of the two-stage explicit Runge– Kutta methods, with the property of having an A-stability region that varies during the integration in accordance with the accuracy requirements. Some numerical experiments on classical stiff problems are presented.
{"title":"A class of explicit one-step methods of order two for stiff problems","authors":"P. Novati","doi":"10.1515/156939505774286120","DOIUrl":"https://doi.org/10.1515/156939505774286120","url":null,"abstract":"In this paper we introduce a new class of explicit one-step methods of order 2 that can be used for solving stiff problems. This class constitutes a generalization of the two-stage explicit Runge– Kutta methods, with the property of having an A-stability region that varies during the integration in accordance with the accuracy requirements. Some numerical experiments on classical stiff problems are presented.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121967602","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}
Pub Date : 2005-06-01DOI: 10.1515/1569395054012785
Y. Efendiev, A. Pankov
In this paper we obtain Meyers type L p+ε-estimates for approximate solutions of nonlinear parabolic equations. This research is motivated by a numerical homogenization of these type of equations [2]. Using derived estimates we show the convergence of numerical solutions obtained from numerical homogenization methods.
{"title":"Meyers type estimates for approximate solutions of nonlinear parabolic equations and their applications","authors":"Y. Efendiev, A. Pankov","doi":"10.1515/1569395054012785","DOIUrl":"https://doi.org/10.1515/1569395054012785","url":null,"abstract":"In this paper we obtain Meyers type L p+ε-estimates for approximate solutions of nonlinear parabolic equations. This research is motivated by a numerical homogenization of these type of equations [2]. Using derived estimates we show the convergence of numerical solutions obtained from numerical homogenization methods.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129507422","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}
Pub Date : 2005-06-01DOI: 10.1515/1569395054012767
W. Hackbusch, Boris N. Khoromskij, E. Tyrtyshnikov
The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integro-differential) operators, especially for high-dimensional problems. These new formats elaborate on a sum of few terms of Kronecker products of smaller-sized matrices (cf. [37,38]). In addition to this we need that the Kronecker factors possess a certain data-sparse structure. Depending on the construction of the Kronecker factors we are led to so-called 'profile-low-rank matrices' or hierarchical matrices (cf. [18,19]). We give a proof for the existence of such formats and expound a gainful combination of the Kronecker-tensor-product structure and the arithmetic for hierarchical matrices.
{"title":"Hierarchical Kronecker tensor-product approximations","authors":"W. Hackbusch, Boris N. Khoromskij, E. Tyrtyshnikov","doi":"10.1515/1569395054012767","DOIUrl":"https://doi.org/10.1515/1569395054012767","url":null,"abstract":"The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integro-differential) operators, especially for high-dimensional problems. These new formats elaborate on a sum of few terms of Kronecker products of smaller-sized matrices (cf. [37,38]). In addition to this we need that the Kronecker factors possess a certain data-sparse structure. Depending on the construction of the Kronecker factors we are led to so-called 'profile-low-rank matrices' or hierarchical matrices (cf. [18,19]). We give a proof for the existence of such formats and expound a gainful combination of the Kronecker-tensor-product structure and the arithmetic for hierarchical matrices.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130029742","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}
Pub Date : 2005-04-01DOI: 10.1515/1569395054068991
F. Suttmeier
The techniques to derive residual based error estimators for finite element discretisations of variational equations can be extended directly to variational inequalities by employing a suitable adaptation of Nitsche's idea (c.f. [8]). This strategy is presented here for elliptic variational inequalities. Its application is demonstrated at the obstacle problem, where numerical results show that the proposed approach to a posteriori error control gives useful error bounds.
{"title":"On a direct approach to adaptive FE-discretisations for elliptic variational inequalities","authors":"F. Suttmeier","doi":"10.1515/1569395054068991","DOIUrl":"https://doi.org/10.1515/1569395054068991","url":null,"abstract":"The techniques to derive residual based error estimators for finite element discretisations of variational equations can be extended directly to variational inequalities by employing a suitable adaptation of Nitsche's idea (c.f. [8]). This strategy is presented here for elliptic variational inequalities. Its application is demonstrated at the obstacle problem, where numerical results show that the proposed approach to a posteriori error control gives useful error bounds.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"148 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123210475","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}
Pub Date : 2005-04-01DOI: 10.1515/1569395054068973
Y. Kuznetsov, S. Repin
In [2] we introduced a new type of mixed finite element approximations for two- and three-dimensional problems on distorted polygonal and polyhedral meshes that consist of cells having different forms. Additional degrees of freedom that arise in the process are excluded by a special condition that is natural for the mixed finite element approximations considered. This paper is devoted to the error analysis of the respective finite element solutions. We show that under certain assumptions on the regularity of the exact solution the convergence rate for the new approximations is the same as for the Raviart–Thomas finite element approximations of the lowest order.
{"title":"Convergence analysis and error estimates for mixed finite element method on distorted meshes","authors":"Y. Kuznetsov, S. Repin","doi":"10.1515/1569395054068973","DOIUrl":"https://doi.org/10.1515/1569395054068973","url":null,"abstract":"In [2] we introduced a new type of mixed finite element approximations for two- and three-dimensional problems on distorted polygonal and polyhedral meshes that consist of cells having different forms. Additional degrees of freedom that arise in the process are excluded by a special condition that is natural for the mixed finite element approximations considered. This paper is devoted to the error analysis of the respective finite element solutions. We show that under certain assumptions on the regularity of the exact solution the convergence rate for the new approximations is the same as for the Raviart–Thomas finite element approximations of the lowest order.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132050929","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}
Pub Date : 2005-04-01DOI: 10.1515/1569395054068982
I. Angelova, L. Vulkov
High-order finite difference approximations of the solution and the flux to model interface problems in one-dimension are constructed and analyzed. Explicit formulas based on new Marchuk integral identities that give O(h 2), O(h 4),… accuracy are derived. Numerical integration procedures using Lobatto quadratures for computing three-point schemes of any prescribed order of accuracy are developed. A rigorous rate of convergence analysis is presented. Numerical experiments confirm the theoretical results.
{"title":"High-order difference schemes based on new Marchuk integral identities for one-dimensional interface problems","authors":"I. Angelova, L. Vulkov","doi":"10.1515/1569395054068982","DOIUrl":"https://doi.org/10.1515/1569395054068982","url":null,"abstract":"High-order finite difference approximations of the solution and the flux to model interface problems in one-dimension are constructed and analyzed. Explicit formulas based on new Marchuk integral identities that give O(h 2), O(h 4),… accuracy are derived. Numerical integration procedures using Lobatto quadratures for computing three-point schemes of any prescribed order of accuracy are developed. A rigorous rate of convergence analysis is presented. Numerical experiments confirm the theoretical results.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"201202 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132441564","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}
Pub Date : 2005-04-01DOI: 10.1515/1569395054069017
C. Carstensen, R. Hoppe
For the 2D eddy currents equations, we design an adaptive edge finite element method (AEFEM) that guarantees an error reduction of the global discretization error in the H (curl)-norm and thus establishes convergence of the adaptive scheme. The error reduction property relies on a residual-type a posteriori error estimator and is proved for discretizations based on the lowest order edge elements of Nédélec's first family. The main ingredients of the proof are the reliability and the strict discrete local efficiency of the estimator as well as the Galerkin orthogonality of the edge element approximation.
{"title":"Convergence analysis of an adaptive edge finite element method for the 2D eddy current equations","authors":"C. Carstensen, R. Hoppe","doi":"10.1515/1569395054069017","DOIUrl":"https://doi.org/10.1515/1569395054069017","url":null,"abstract":"For the 2D eddy currents equations, we design an adaptive edge finite element method (AEFEM) that guarantees an error reduction of the global discretization error in the H (curl)-norm and thus establishes convergence of the adaptive scheme. The error reduction property relies on a residual-type a posteriori error estimator and is proved for discretizations based on the lowest order edge elements of Nédélec's first family. The main ingredients of the proof are the reliability and the strict discrete local efficiency of the estimator as well as the Galerkin orthogonality of the edge element approximation.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114447487","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}
Pub Date : 2005-04-01DOI: 10.1515/1569395054069008
Zhenjun Shi
The paper presents a new class of memory gradient methods with inexact line searches for unconstrained minimization problems. The methods use more previous iterative information than other methods to generate a search direction and use inexact line searches to select a step-size at each iteration. It is proved that the new methods have global convergence under weak mild conditions. The convergence rate of these methods is also investigated under some special cases. Some numerical experiments show that these new algorithms converge more stably than other line search methods and are effective in solving large scale unconstrained minimization problems.
{"title":"A new class of memory gradient methods with inexact line searches","authors":"Zhenjun Shi","doi":"10.1515/1569395054069008","DOIUrl":"https://doi.org/10.1515/1569395054069008","url":null,"abstract":"The paper presents a new class of memory gradient methods with inexact line searches for unconstrained minimization problems. The methods use more previous iterative information than other methods to generate a search direction and use inexact line searches to select a step-size at each iteration. It is proved that the new methods have global convergence under weak mild conditions. The convergence rate of these methods is also investigated under some special cases. Some numerical experiments show that these new algorithms converge more stably than other line search methods and are effective in solving large scale unconstrained minimization problems.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"36 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124002890","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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