Maximum Norm A Posteriori Error Estimates for a One-Dimensional Convection-Diffusion Problem

Kopteva, Natalia

doi:10.1137/s0036142900368642

Cited by 61 publications

(39 citation statements)

References 18 publications

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“…For the terminology, we refer to [25] and [23]. Here we follow [1,25] to use the discrete Green functions.…”

Section: Stability Of the Sdfemmentioning

confidence: 99%

“…Basically we have two ways to prove the convergence of a fully adapted algorithm. One is to derive a posteriori error estimate for our optimal SDFEM using the framework in [23]. It is interesting to note that the estimate (4.4) in [23] for a different second-order method is a discrete version of the estimate given by Theorem 5.3.…”

Section: Conclusion and Further Remarksmentioning

confidence: 99%

“…One is to derive a posteriori error estimate for our optimal SDFEM using the framework in [23]. It is interesting to note that the estimate (4.4) in [23] for a different second-order method is a discrete version of the estimate given by Theorem 5.3. Another approach is to recovery u from the numerical solution, in which the superconvergence theory will play an important role.…”

Section: Conclusion and Further Remarksmentioning

confidence: 99%

See 2 more Smart Citations

An optimal streamline diffusion finite element method for a singularly perturbed problem

Chen¹,

Xu²

2005

Recent Advances in Adaptive Computation

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Abstract. The stability and accuracy of a streamline diffusion finite element method (SDFEM) on arbitrary grids applied to a linear 1-d singularly perturbed problem are studied in this paper. With a special choice of the stabilization quadratic bubble function, the SDFEM is shown to have an optimal second order in the sense thatwhere u h is the SDFEM approximation of the exact solution u and V h is the linear finite element space. With the quasi-optimal interpolation error estimate, quasi-optimal convergence results for the SDFEM are obtained. As a consequence, an open question about the optimal choice of the monitor function for a second order scheme in the moving mesh method is answered.

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“…For the terminology, we refer to [25] and [23]. Here we follow [1,25] to use the discrete Green functions.…”

Section: Stability Of the Sdfemmentioning

confidence: 99%

Section: Conclusion and Further Remarksmentioning

confidence: 99%

Section: Conclusion and Further Remarksmentioning

confidence: 99%

See 1 more Smart Citation

An optimal streamline diffusion finite element method for a singularly perturbed problem

Chen¹,

Xu²

2005

Recent Advances in Adaptive Computation

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“…It could be done by a posterior error estimate [34] or a priori estimate of u . For example, let us consider the following equation as studied in [48].…”

Section: Corollarymentioning

confidence: 99%

Stability and accuracy of adapted finite element methods for singularly perturbed problems

Chen

2008

Numer. Math.

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The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sensewhere u N is the SDFEM approximation in linear finite element space V N of the exact solution u. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered. Mathematics Subject Classification (2000)65L10 · 65L20 · 65L60 · 76R99

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“…(The few known a posteriori error estimates for anisotropic meshes are in a weaker energy norm; see, e.g., [21,22].) We also refer the reader to related papers on maximum norm a posteriori error estimation in one dimension [16,17,20,24].…”

Section: Introductionmentioning

confidence: 99%

Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem

Chadha

Kopteva

2010

Adv Comput Math

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Abstract. A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green's function of the continuous differential operator in the Sobolev W 1,1 and W 2,1 norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate.

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Maximum Norm A Posteriori Error Estimates for a One-Dimensional Convection-Diffusion Problem

Cited by 61 publications

References 18 publications

An optimal streamline diffusion finite element method for a singularly perturbed problem

An optimal streamline diffusion finite element method for a singularly perturbed problem

Stability and accuracy of adapted finite element methods for singularly perturbed problems

Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem

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