Pub Date : 2004-06-01DOI: 10.1515/156939504323074496
Hongsen Chen
This paper derives some interior superconvergence estimates for finite element solutions of the Stokes problem by a local L 2 projection method. The results depend only on local properties of the Stokes problem and the finite element approximations.
{"title":"Interior superconvergence of finite element solutions for Stokes problems by local L 2 projections","authors":"Hongsen Chen","doi":"10.1515/156939504323074496","DOIUrl":"https://doi.org/10.1515/156939504323074496","url":null,"abstract":"This paper derives some interior superconvergence estimates for finite element solutions of the Stokes problem by a local L 2 projection method. The results depend only on local properties of the Stokes problem and the finite element approximations.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123498914","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 : 2004-06-01DOI: 10.1515/156939504323074522
P. Moore
Hp-adaptive finite element methods require error estimates of the solution at the current order and one order higher. Hierarchical-based estimation strategies have proved effective in computing errors at the current order for nonlinear parabolic equations. Recently a new approach, interpolation error-based (IEB) error estimation, for constructing a posteriori error estimates at both orders has been developed for linear reaction-diffusion equations. The main results are: (i) IEB error estimation can be applied to nonlinear reaction-diffusion equations in one space dimension; (ii) the hierarchical estimator is an implicit IEB method and thus, works for reaction-diffusion problems; (iii) a hierarchical extension for computing higher–order error estimates is asymptotically exact. Computational results illustrating the theory and comparing the implicit (hierarchical) strategy with the earlier explicit IEB methods are presented.
{"title":"An implicit interpolation error-based error estimation strategy for hp-adaptivity in one space dimension","authors":"P. Moore","doi":"10.1515/156939504323074522","DOIUrl":"https://doi.org/10.1515/156939504323074522","url":null,"abstract":"Hp-adaptive finite element methods require error estimates of the solution at the current order and one order higher. Hierarchical-based estimation strategies have proved effective in computing errors at the current order for nonlinear parabolic equations. Recently a new approach, interpolation error-based (IEB) error estimation, for constructing a posteriori error estimates at both orders has been developed for linear reaction-diffusion equations. The main results are: (i) IEB error estimation can be applied to nonlinear reaction-diffusion equations in one space dimension; (ii) the hierarchical estimator is an implicit IEB method and thus, works for reaction-diffusion problems; (iii) a hierarchical extension for computing higher–order error estimates is asymptotically exact. Computational results illustrating the theory and comparing the implicit (hierarchical) strategy with the earlier explicit IEB methods are presented.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117117560","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 : 2004-04-01DOI: 10.1515/1569395041172917
M. Dryja, A. Gantner, O. Widlund, B. Wohlmuth
Mortar elements form a family of special non-overlapping domain decomposition methods which allows the coupling of different triangulations across subdomain boundaries. We discuss and analyze a multilevel preconditioner for mortar finite elements on nonmatching triangulations. The analysis is carried out within the abstract framework of additive Schwarz methods. Numerical results show a performance of our preconditioner as predicted by the theory. Our condition number estimate depends quadratically on the number of refinement levels.
{"title":"Multilevel additive Schwarz preconditioner for nonconforming mortar finite element methods","authors":"M. Dryja, A. Gantner, O. Widlund, B. Wohlmuth","doi":"10.1515/1569395041172917","DOIUrl":"https://doi.org/10.1515/1569395041172917","url":null,"abstract":"Mortar elements form a family of special non-overlapping domain decomposition methods which allows the coupling of different triangulations across subdomain boundaries. We discuss and analyze a multilevel preconditioner for mortar finite elements on nonmatching triangulations. The analysis is carried out within the abstract framework of additive Schwarz methods. Numerical results show a performance of our preconditioner as predicted by the theory. Our condition number estimate depends quadratically on the number of refinement levels.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124816717","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 : 2004-04-01DOI: 10.1515/1569395041172944
Jichun Li
The standard conforming finite element methods on one type of highly nonuniform rectangular meshes are considered for solving the quasilinear singular perturbation problem -ε2(u xx + u yy ) + ƒ(x,y;u) = 0. By using a special interpolation operator and the integral identity technique, optimal uniform convergence rates of O(N –(k+1)) in the L2-norm are obtained for all k-th (k ≥ 1) order conforming tensor-product finite elements, where N is the number of intervals in both x- and y-directions. Hence Apel and Lube's suboptimal results are improved to optimal order and generalized to the quasilinear case.
针对拟线性奇异摄动问题ε2(u xx + u yy) + f (x,y;u) = 0，考虑了一类高度不均匀矩形网格的标准拟合有限元方法。利用一种特殊的插值算子和积分恒等技术，得到了所有k (k≥1)阶符合张量积有限元在l2范数上O(N - (k+1))的最优一致收敛速率，其中N为x方向和y方向上的区间数。将Apel和Lube的次优结果改进到最优阶，并推广到拟线性情况。
{"title":"Optimal uniform convergence analysis for a singularly perturbed quasilinear reaction–diffusion problem","authors":"Jichun Li","doi":"10.1515/1569395041172944","DOIUrl":"https://doi.org/10.1515/1569395041172944","url":null,"abstract":"The standard conforming finite element methods on one type of highly nonuniform rectangular meshes are considered for solving the quasilinear singular perturbation problem -ε2(u xx + u yy ) + ƒ(x,y;u) = 0. By using a special interpolation operator and the integral identity technique, optimal uniform convergence rates of O(N –(k+1)) in the L2-norm are obtained for all k-th (k ≥ 1) order conforming tensor-product finite elements, where N is the number of intervals in both x- and y-directions. Hence Apel and Lube's suboptimal results are improved to optimal order and generalized to the quasilinear case.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117122568","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 : 2004-04-01DOI: 10.1515/1569395041172926
Hongsen Chen
In this paper local error estimates for mixed discontinuous Galerkin methods including the local discontinuous Galerkin method for solving second-order elliptic problems are established. Our result shows that the errors of both the vector and scalar solutions of the mixed DG methods in a local subdomain are bounded by the local approximation properties of the finite element spaces plus the errors measured in the negative Sobolev norms in a slightly larger subdomain.
{"title":"Local error estimates of mixed discontinuous Galerkin methods for elliptic problems","authors":"Hongsen Chen","doi":"10.1515/1569395041172926","DOIUrl":"https://doi.org/10.1515/1569395041172926","url":null,"abstract":"In this paper local error estimates for mixed discontinuous Galerkin methods including the local discontinuous Galerkin method for solving second-order elliptic problems are established. Our result shows that the errors of both the vector and scalar solutions of the mixed DG methods in a local subdomain are bounded by the local approximation properties of the finite element spaces plus the errors measured in the negative Sobolev norms in a slightly larger subdomain.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"349 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122765038","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 : 2004-04-01DOI: 10.1515/1569395041172935
M. Stolper
The boundary integral formulation for the Dirichlet boundary value problem is considered and the collocation boundary element method for the discretisation of the problem is used. In order to compute the entries of the matrices for several wave numbers, the inverse Fourier transform with respect to the wave number is applied to them. The analytical forms and some important properties of the transformed matrices are deduced. After applying the Fourier transform, new matrices depending on the wave number are obtained and the associated linear systems are treated. Further, the adaptive cross approximation (ACA) algorithm is applied to the matrices solving efficiently the linear systems. Finally, some numerical examples for the solution are presented.
{"title":"Computing and compression of the boundary element matrices for the Helmholtz equation","authors":"M. Stolper","doi":"10.1515/1569395041172935","DOIUrl":"https://doi.org/10.1515/1569395041172935","url":null,"abstract":"The boundary integral formulation for the Dirichlet boundary value problem is considered and the collocation boundary element method for the discretisation of the problem is used. In order to compute the entries of the matrices for several wave numbers, the inverse Fourier transform with respect to the wave number is applied to them. The analytical forms and some important properties of the transformed matrices are deduced. After applying the Fourier transform, new matrices depending on the wave number are obtained and the associated linear systems are treated. Further, the adaptive cross approximation (ACA) algorithm is applied to the matrices solving efficiently the linear systems. Finally, some numerical examples for the solution are presented.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116815638","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 : 2003-12-01DOI: 10.1515/156939503322663467
L. Lefton, Dongming Wei
Finite element approximations of the stationary power-law Stokes problem using penalty formulation are considered. A priori error estimates under appropriate smoothness assumptions on the solutions are established without assuming a discrete version of the BB condition. Numerical solutions are presented by implementing a nonlinear conjugate gradient method.
{"title":"Penalty finite element approximations of the stationary power-law Stokes problem","authors":"L. Lefton, Dongming Wei","doi":"10.1515/156939503322663467","DOIUrl":"https://doi.org/10.1515/156939503322663467","url":null,"abstract":"Finite element approximations of the stationary power-law Stokes problem using penalty formulation are considered. A priori error estimates under appropriate smoothness assumptions on the solutions are established without assuming a discrete version of the BB condition. Numerical solutions are presented by implementing a nonlinear conjugate gradient method.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130187968","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 : 2003-12-01DOI: 10.1515/156939503322663449
Hongsen Chen, Zhangxin Chen
In this paper we develop an abstract theory for stability and convergence of mixed discontinuous finite element methods for second-order partial differential problems. This theory is then applied to various examples, with an emphasis on different combinations of mixed finite element spaces. Elliptic, parabolic, and convection-dominated diffusion problems are considered. The examples include classical mixed finite element methods in the discontinuous setting, local discontinuous Galerkin methods, and their penalized (stablized) versions. For the convection-dominated diffusion problems, a characteristics-based approach is combined with the mixed discontinuous methods.
{"title":"Stability and convergence of mixed discontinuous finite element methods for second-order differential problems","authors":"Hongsen Chen, Zhangxin Chen","doi":"10.1515/156939503322663449","DOIUrl":"https://doi.org/10.1515/156939503322663449","url":null,"abstract":"In this paper we develop an abstract theory for stability and convergence of mixed discontinuous finite element methods for second-order partial differential problems. This theory is then applied to various examples, with an emphasis on different combinations of mixed finite element spaces. Elliptic, parabolic, and convection-dominated diffusion problems are considered. The examples include classical mixed finite element methods in the discontinuous setting, local discontinuous Galerkin methods, and their penalized (stablized) versions. For the convection-dominated diffusion problems, a characteristics-based approach is combined with the mixed discontinuous methods.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125251128","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 : 2003-09-01DOI: 10.1515/156939503322553090
A. Knyazev
We concentrate on a model diffusion equation on a Lipschitz simply connected bounded domain with a small diffusion coefficient in a Lipschitz simply connected subdomain located strictly inside of the original domain. We study asymptotic properties of the solution with respect to the small diffusion coefficient vanishing. It is known that the solution asymptotically turns into a solution of a corresponding diffusion equation with Neumann boundary conditions on a part of the boundary. One typical proof technique of this fact utilizes a reduction of the problem to the interface of the subdomain, using a transmission condition. An analogous approach appears in studying domain decomposition methods without overlap, reducing the investigation to the surface that separates the subdomains and in theoretical foundation of a fictitious domain, also called embedding, method, e.g., to prove a classical estimate that guaranties convergence of the solution of the fictitious domain problem to the solution of the original Neumann boundary value problem. On a continuous level, this analysis is usually performed in an H 1/2 norm for second order elliptic equations. This norm appears naturally for Poincaré-Steklov operators, which are convenient to employ to formulate the transmission condition. Using recent advances in regularity theory of Poincaré-Steklov operators for Lipschitz domains, we provide, in the present paper, a similar analysis in an H 1/2+α norm with α > 0, for a simple model problem. This result leads to a convergence theory of the fictitious domain method for a second order elliptic PDE in an H 1+α norm, while the classical result is in an H 1 norm. Here, α < 1/2 for the case of Lipschitz domains we consider.
{"title":"Analysis of transmission problems on Lipschitz boundaries in stronger norms","authors":"A. Knyazev","doi":"10.1515/156939503322553090","DOIUrl":"https://doi.org/10.1515/156939503322553090","url":null,"abstract":"We concentrate on a model diffusion equation on a Lipschitz simply connected bounded domain with a small diffusion coefficient in a Lipschitz simply connected subdomain located strictly inside of the original domain. We study asymptotic properties of the solution with respect to the small diffusion coefficient vanishing. It is known that the solution asymptotically turns into a solution of a corresponding diffusion equation with Neumann boundary conditions on a part of the boundary. One typical proof technique of this fact utilizes a reduction of the problem to the interface of the subdomain, using a transmission condition. An analogous approach appears in studying domain decomposition methods without overlap, reducing the investigation to the surface that separates the subdomains and in theoretical foundation of a fictitious domain, also called embedding, method, e.g., to prove a classical estimate that guaranties convergence of the solution of the fictitious domain problem to the solution of the original Neumann boundary value problem. On a continuous level, this analysis is usually performed in an H 1/2 norm for second order elliptic equations. This norm appears naturally for Poincaré-Steklov operators, which are convenient to employ to formulate the transmission condition. Using recent advances in regularity theory of Poincaré-Steklov operators for Lipschitz domains, we provide, in the present paper, a similar analysis in an H 1/2+α norm with α > 0, for a simple model problem. This result leads to a convergence theory of the fictitious domain method for a second order elliptic PDE in an H 1+α norm, while the classical result is in an H 1 norm. Here, α < 1/2 for the case of Lipschitz domains we consider.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124928717","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 : 2003-09-01DOI: 10.1515/156939503322553081
B. Khoromskij
In the preceding paper [24], a method is described for an explicit hierarchical (ℋ-matrix) approximation to the inverse of an elliptic differential operator with piecewise constant/smooth coefficients in ℝ d . In the present paper, we proceed with the ℋ-matrix approximation to the Green function. Here, it is represented by a sum of an ℋ-matrix and certain correction term including the product of data-sparse matrices of hierarchical formats based on the so-called boundary concentrated FEM [26]. In the case of jumping coefficients with respect to non-overlapping domain decomposition, the approximate inverse operator is obtained as a direct sum of local inverses over subdomains and the Schur complement inverse on the interface corresponding to the boundary concentrated FEM. Our Schur complement matrix provides the cheap spectrally equivalent preconditioner to the conventional interface operator arising in the iterative substructuring methods by piecewise linear finite elements.
{"title":"Hierarchical matrix approximation to Green's function via boundary concentrated FEM","authors":"B. Khoromskij","doi":"10.1515/156939503322553081","DOIUrl":"https://doi.org/10.1515/156939503322553081","url":null,"abstract":"In the preceding paper [24], a method is described for an explicit hierarchical (ℋ-matrix) approximation to the inverse of an elliptic differential operator with piecewise constant/smooth coefficients in ℝ d . In the present paper, we proceed with the ℋ-matrix approximation to the Green function. Here, it is represented by a sum of an ℋ-matrix and certain correction term including the product of data-sparse matrices of hierarchical formats based on the so-called boundary concentrated FEM [26]. In the case of jumping coefficients with respect to non-overlapping domain decomposition, the approximate inverse operator is obtained as a direct sum of local inverses over subdomains and the Schur complement inverse on the interface corresponding to the boundary concentrated FEM. Our Schur complement matrix provides the cheap spectrally equivalent preconditioner to the conventional interface operator arising in the iterative substructuring methods by piecewise linear finite elements.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127311988","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：