Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities

Franz, Sebastian; Kellogg, R. Bruce; Stynes, Martin

doi:10.1090/s0025-5718-2011-02526-3

Cited by 15 publications

(8 citation statements)

References 13 publications

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“…Observe that since ϕ h (x J−1 ) = 0 for ϕ h not proportional to ϕ J−1 , and recalling how we defined α * in (14), equation (16) is similar to (7)(8), and then, the leastsquares problem (15) is the way of finding the value α hopefully close to α * . Notice also that the restriction (16) is, in fact, a set of as many independent restrictions as interior nodes or nodal basis functions in V h .…”

Section: The One-dimensional Casementioning

confidence: 99%

“…Among these we cite Shishkin meshes (described below) [31], [37], which have received considerable attention in recent years [14], [15], [16], [28], [32], [33] [39], [43] and [47]. However, it is generally acknowledged that the main drawback of Shishkin meshes is the difficulty to design them on domains with nontrivial geometries, although some works overcoming this difficulty can be found in the literature [46], [28].…”

Section: Introductionmentioning

confidence: 99%

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Shishkin mesh simulation: A new stabilization technique for convection–diffusion problems

García-Archilla

2013

Computer Methods in Applied Mechanics and Engineering

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We present together in this document the paper B. García-Archilla, Shiskin mesh simulation: A new stabilization technique for convectiondiffusion problems, Comput. Methods Appl. Mech. Engrg., 256 (2013), 1-16, and the manuscript B. García-Archilla, Shiskin mesh simulation: Complementary experiments, which contains some of the experiments that, for the sake of brevity were not included in the first one. This second paper with complementary experiments is neither self-contained nor intended to be published in any major journal , but is intended to be accessible to anyone wishing to learn more on the performance of the SMS method. Following a principle of reproducible computational research, the source codes of the experiments in the present papers are available from the author on request or on "http://personal.us.es/bga/bga files/software bga.html". Excluded are the codes corresponding to Example 7 which is subject of current resarch * Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain. AbstractA new stabilization procedure is presented. It is based on a simulation of the interaction between the coarse and fine parts of a Shishkin mesh, but can be applied on coarse and irregular meshes and on domains with nontrivial geometries. The technique, which does not require adjusting any parameter, can be applied to different stabilized and non stabilized methods. Numerical experiments show it to obtain oscillation-free approximations on problems with boundary and internal layers, on uniform and nonuniform meshes and on domains with curved boundaries.

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Section: The One-dimensional Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Shishkin mesh simulation: A new stabilization technique for convection–diffusion problems

García-Archilla

2013

Computer Methods in Applied Mechanics and Engineering

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“…For time‐dependent problems, we use the SD method discrete only in space variables and the finite difference discrete in time direction to derive the finite difference streamline diffusion (FDSD) method . This method keeps the essential aspect of the original SD method and simplifies the algorithm structure . However, the SD/FDSD methods have some undesirable features: introducing additional nonphysical coupling terms between velocity and pressure; producing inaccurate numerical solutions near the boundary; having to calculate second derivative when using high order elements.…”

Section: Introductionmentioning

confidence: 99%

Local projection stabilized and characteristic decoupled scheme for the fluid–fluid interaction problems

Qian

Chen

Feng

2016

Numerical Methods Partial

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In this article, we propose and analyse a local projection stabilized and characteristic decoupled scheme for the fluid–fluid interaction problems. We use the method of characteristics type to avert the difficulties caused by the nonlinear term, and use the local projection stabilized method to control spurious oscillations in the velocities due to dominant convection, and use a geometric averaging idea to decouple the monolithic problems. The stability analysis is derived and numerical tests are performed to demonstrate the robustness of this new method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 704–723, 2017

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“…Among them, the streamline diffusion finite element method (SDFEM) [7] combined with the Shishkin mesh [14] presents good numerical performances and has been widely studied, see [17,5,3,18].…”

Section: Introductionmentioning

confidence: 99%

Analysis of SDFEM on Shishkin Triangular Meshes and Hybrid Meshes for Problems with Characteristic Layers

Zhang

Liu

2016

J Sci Comput

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In this paper, we analyze the streamline diffusion finite element method (SDFEM) for a model singularly perturbed convection-diffusion equation on a Shishkin triangular mesh and hybrid meshes. Supercloseness property of u I − u N is obtained, where u I is the interpolant of the solution u and u N is the SDFEM's solution. The analysis depends on novel integral inequalities for the diffusion and convection parts in the bilinear form. Furthermore, analysis on hybrid meshes shows that bilinear elements should be recommended for the exponential layer, not for the characteristic layer. Finally, numerical experiments support these theoretical results.

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Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities

Cited by 15 publications

References 13 publications

Shishkin mesh simulation: A new stabilization technique for convection–diffusion problems

Shishkin mesh simulation: A new stabilization technique for convection–diffusion problems

Local projection stabilized and characteristic decoupled scheme for the fluid–fluid interaction problems

Analysis of SDFEM on Shishkin Triangular Meshes and Hybrid Meshes for Problems with Characteristic Layers

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