This paper is on the efficient solution of linear systems arising in discretizations of second order elliptic PDEs by a generalized finite element method (GFEM). Our results apply for GFEM equations on unstructured simplicial grids in 2 and 3 spatial dimensions. We propose an efficient preconditioner by using auxiliary (fictitious) space techniques and an additive preconditioner for the auxiliary space problems. We also prove that the condition number of the preconditioned system is uniformly bounded with respect to the mesh parameters.
{"title":"A multilevel preconditioning for generalized finite element method problems on unstructured simplicial meshes","authors":"D. Cho, L. Zikatanov","doi":"10.1515/jnma.2007.008","DOIUrl":"https://doi.org/10.1515/jnma.2007.008","url":null,"abstract":"This paper is on the efficient solution of linear systems arising in discretizations of second order elliptic PDEs by a generalized finite element method (GFEM). Our results apply for GFEM equations on unstructured simplicial grids in 2 and 3 spatial dimensions. We propose an efficient preconditioner by using auxiliary (fictitious) space techniques and an additive preconditioner for the auxiliary space problems. We also prove that the condition number of the preconditioned system is uniformly bounded with respect to the mesh parameters.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129880288","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}
In this paper we prove the discrete maximum principle for a one-dimensional equation of the form –(au′)′ = f with piecewise-constant coefficient a(x), discretized by the hp-FEM. The discrete problem is transformed in such a way that the discontinuity of the coefficient a(x) disappears. Existing results are then applied to obtain a condition on the mesh which guarantees the satisfaction of the discrete maximum principle. Both Dirichlet and mixed Dirichlet–Neumann boundary conditions are discussed.
{"title":"Discrete maximum principle for a 1D problem with piecewise-constant coefficients solved by hp-FEM","authors":"T. Vejchodský, P. Solín","doi":"10.1515/jnma.2007.011","DOIUrl":"https://doi.org/10.1515/jnma.2007.011","url":null,"abstract":"In this paper we prove the discrete maximum principle for a one-dimensional equation of the form –(au′)′ = f with piecewise-constant coefficient a(x), discretized by the hp-FEM. The discrete problem is transformed in such a way that the discontinuity of the coefficient a(x) disappears. Existing results are then applied to obtain a condition on the mesh which guarantees the satisfaction of the discrete maximum principle. Both Dirichlet and mixed Dirichlet–Neumann boundary conditions are discussed.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"74 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131613501","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 : 2006-12-01DOI: 10.1515/156939506779874617
Y. Kuznetsov
In this paper, a new mixed finite element method for the diffusion equation on polygonal meshes is proposed. The method is applied to the diffusion equation on meshes with mixed cells when all the coefficients and the source function may have discontinuities inside polygonal mesh cells. The resulting discrete equations operate only with the degrees of freedom for normal fluxes on the boundaries of cells and one degree of freedom per cell for the solution function.
{"title":"Mixed finite element method for diffusion equations on polygonal meshes with mixed cells","authors":"Y. Kuznetsov","doi":"10.1515/156939506779874617","DOIUrl":"https://doi.org/10.1515/156939506779874617","url":null,"abstract":"In this paper, a new mixed finite element method for the diffusion equation on polygonal meshes is proposed. The method is applied to the diffusion equation on meshes with mixed cells when all the coefficients and the source function may have discontinuities inside polygonal mesh cells. The resulting discrete equations operate only with the degrees of freedom for normal fluxes on the boundaries of cells and one degree of freedom per cell for the solution function.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129769261","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 : 2006-12-01DOI: 10.1515/156939506779874626
Stefan Kunis, D. Potts, G. Steidl
We construct a fast algorithm for the computation of discrete Gauss transforms with complex parameters, capable of dealing with non equispaced points. Our algorithm is based on the fast Fourier transform at non equispaced knots and requires only (N) arithmetic operations.
{"title":"Fast Gauss transforms with complex parameters using NFFTs","authors":"Stefan Kunis, D. Potts, G. Steidl","doi":"10.1515/156939506779874626","DOIUrl":"https://doi.org/10.1515/156939506779874626","url":null,"abstract":"We construct a fast algorithm for the computation of discrete Gauss transforms with complex parameters, capable of dealing with non equispaced points. Our algorithm is based on the fast Fourier transform at non equispaced knots and requires only (N) arithmetic operations.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116467929","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 : 2006-12-01DOI: 10.1515/156939506779874608
R. Ciarski, Z. Kamont
Nonlinear first order partial functional differential systems are considered in the paper. Classical solutions of the local Cauchy problem on the Haar pyramid are approximated by solutions of suitable quasilinear systems of difference functional equations. The proposed numerical methods are difference schemes of the Euler type. A complete convergence analysis is given and it is shown by examples that the new methods are considerable better than the classical methods. It is shown that the Lax scheme is superfluous for the numerical approximations of classical solutions to nonlinear functional differential problems. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators.
{"title":"Generalized Euler method for Hamilton Jacobi differential functional systems","authors":"R. Ciarski, Z. Kamont","doi":"10.1515/156939506779874608","DOIUrl":"https://doi.org/10.1515/156939506779874608","url":null,"abstract":"Nonlinear first order partial functional differential systems are considered in the paper. Classical solutions of the local Cauchy problem on the Haar pyramid are approximated by solutions of suitable quasilinear systems of difference functional equations. The proposed numerical methods are difference schemes of the Euler type. A complete convergence analysis is given and it is shown by examples that the new methods are considerable better than the classical methods. It is shown that the Lax scheme is superfluous for the numerical approximations of classical solutions to nonlinear functional differential problems. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"88 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116657559","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 : 2006-12-01DOI: 10.1515/156939506779874590
I. Boglaev
This paper deals with monotone iterative algorithms for solving nonlinear monotone difference schemes of parabolic type. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing the nonlinear monotone difference schemes in the canonical form. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear monotone difference schemes. Numerical experiments are presented.
{"title":"Monotone algorithms for solving nonlinear monotone difference schemes of parabolic type in the canonical form","authors":"I. Boglaev","doi":"10.1515/156939506779874590","DOIUrl":"https://doi.org/10.1515/156939506779874590","url":null,"abstract":"This paper deals with monotone iterative algorithms for solving nonlinear monotone difference schemes of parabolic type. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing the nonlinear monotone difference schemes in the canonical form. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear monotone difference schemes. Numerical experiments are presented.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131461336","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 : 2006-09-01DOI: 10.1515/156939506778658311
Markus Jürgens
In this article an inherently parallel algorithm to approximate the operator exponential is presented. The construction is based on the integral representation of the operator exponential and allows arbitrarily large time steps constituting a major advantage compared to classical schemes. The algorithm rests on the efficient solution of several elliptic problems depending on a complex parameter. We prove Besov regularity of the solutions to these elliptic problems. This result implies the efficiency of adaptive methods applied to the elliptic problems and leads to a complexity estimate for the complete algorithm. In the numerical experiments the efficiency of the new scheme is demonstrated by comparison to a single step method of second order.
{"title":"Adaptive application of the operator exponential","authors":"Markus Jürgens","doi":"10.1515/156939506778658311","DOIUrl":"https://doi.org/10.1515/156939506778658311","url":null,"abstract":"In this article an inherently parallel algorithm to approximate the operator exponential is presented. The construction is based on the integral representation of the operator exponential and allows arbitrarily large time steps constituting a major advantage compared to classical schemes. The algorithm rests on the efficient solution of several elliptic problems depending on a complex parameter. We prove Besov regularity of the solutions to these elliptic problems. This result implies the efficiency of adaptive methods applied to the elliptic problems and leads to a complexity estimate for the complete algorithm. In the numerical experiments the efficiency of the new scheme is demonstrated by comparison to a single step method of second order.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129095490","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 : 2006-06-01DOI: 10.1515/156939506777443031
F. Suttmeier
In this note, we continue our studies on optimised mesh design for the Finite Element (FE) method using global norm estimates for local error control. The strategies are based on the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes (see, e.g., [3,7,18]), where error control for the primal problem is established by solving an auxiliary (dual) problem. In this context we blamed (cf. [17,18]) global norm estimates being not that useful in applications. But having a closer look at the DWR-concept, one observes that in fact global error bounds can be employed to establish local error control. We derive rigorous error bounds, especially we control the approximation process of the (unknown) dual solution entering the proposed estimate. Additional, these estimates provide information to optimise the approximation process of the primal and dual problem.
{"title":"On adaptive computational methods: global norms controlling local errors","authors":"F. Suttmeier","doi":"10.1515/156939506777443031","DOIUrl":"https://doi.org/10.1515/156939506777443031","url":null,"abstract":"In this note, we continue our studies on optimised mesh design for the Finite Element (FE) method using global norm estimates for local error control. The strategies are based on the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes (see, e.g., [3,7,18]), where error control for the primal problem is established by solving an auxiliary (dual) problem. In this context we blamed (cf. [17,18]) global norm estimates being not that useful in applications. But having a closer look at the DWR-concept, one observes that in fact global error bounds can be employed to establish local error control. We derive rigorous error bounds, especially we control the approximation process of the (unknown) dual solution entering the proposed estimate. Additional, these estimates provide information to optimise the approximation process of the primal and dual problem.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130878753","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 : 2006-06-01DOI: 10.1515/156939506777443059
K. Mardal, T. Nilssen
In this work we study a preconditioned iterative method for some higher-order time discretizations of linear parabolic partial differential equations. We use the Padé approximations of the exponential function to discretize in time and show that standard solution algorithms for lower-order time discretization schemes, such as Crank–Nicolson and implicit Euler, can be reused as preconditioners for the arising linear system. The proposed preconditioner is order optimal with respect to the discretization parameters.
{"title":"Reuse of standard preconditioners for higher-order time discretizations of parabolic PDEs","authors":"K. Mardal, T. Nilssen","doi":"10.1515/156939506777443059","DOIUrl":"https://doi.org/10.1515/156939506777443059","url":null,"abstract":"In this work we study a preconditioned iterative method for some higher-order time discretizations of linear parabolic partial differential equations. We use the Padé approximations of the exponential function to discretize in time and show that standard solution algorithms for lower-order time discretization schemes, such as Crank–Nicolson and implicit Euler, can be reused as preconditioners for the arising linear system. The proposed preconditioner is order optimal with respect to the discretization parameters.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"187 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117128231","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 : 2006-06-01DOI: 10.1515/156939506777443040
A. Vagharshakyan
In this paper, we investigate a family of numerical methods for the approximate solution of integral equations. Here we shed light on reasons of ill posed effects and investigate several approaches to avoid those problems.
{"title":"On a numerical method for resolution of integral equations","authors":"A. Vagharshakyan","doi":"10.1515/156939506777443040","DOIUrl":"https://doi.org/10.1515/156939506777443040","url":null,"abstract":"In this paper, we investigate a family of numerical methods for the approximate solution of integral equations. Here we shed light on reasons of ill posed effects and investigate several approaches to avoid those problems.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129950444","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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