2002
DOI: 10.1002/num.10001
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Finite element superconvergence approximation for one‐dimensional singularly perturbed problems

Abstract: Superconvergence approximations of singularly perturbed two-point boundary value problems of reactiondiffusion type and convection-diffusion type are studied. By applying the standard finite element method of any fixed order p on a modified Shishkin mesh, superconvergence error bounds of (N −1 ln(N + 1)) p+1 in a discrete energy norm in approximating problems with the exponential type boundary layers are established. The error bounds are uniformly valid with respect to the singular perturbation parameter. Nume… Show more

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Cited by 55 publications
(33 citation statements)
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“…We shall establish an error bound of order N −2 ln 2 N + N −3/2 in the discrete -weighted energy norm under certain regularity assumptions. For the one-dimensional case, see a recent work of the author [24]. As a consequence of the superconvergence result, we obtain convergence of the same order in the L 2 -norm and pointwise convergence of order N −3/2 ln 5/2 N + N −1 ln 1/2 N at some mesh points inside the boundary layer under the same regularity assumption.…”
Section: Introductionsupporting
confidence: 64%
See 1 more Smart Citation
“…We shall establish an error bound of order N −2 ln 2 N + N −3/2 in the discrete -weighted energy norm under certain regularity assumptions. For the one-dimensional case, see a recent work of the author [24]. As a consequence of the superconvergence result, we obtain convergence of the same order in the L 2 -norm and pointwise convergence of order N −3/2 ln 5/2 N + N −1 ln 1/2 N at some mesh points inside the boundary layer under the same regularity assumption.…”
Section: Introductionsupporting
confidence: 64%
“…The reader is referred to three 1996 books [13,14,16] for the significant progress that has been made in this field, and articles [2,4,7,8,11,12,15,18,19,20,21,24,25] for more information.…”
Section: Introductionmentioning
confidence: 99%
“…First we compare the LDG with the standard finite element method (under the Shishkin mesh), which has the following error bound (see [15]) for problem (1.4):…”
Section: Introduction and Statement Of The Main Resultsmentioning
confidence: 99%
“…Another effective way for globally solving this problem is to construct a numerical scheme on a layer adapted mesh, such as Shishkin type meshes and Bakhvalov type meshes. There are plenty of theoretical results about FEMs and stabilized FEMs on layer adapted meshes [6,17,23]. Recently, the LDG method was considered [20,22] for numerically solving singularly perturbed problems on layer adapted meshes.…”
Section: −ǫUmentioning
confidence: 99%