Robust Computational Techniques for Boundary Layers

Farrell, Paul A.; Hegarty, Alan F.; Miller, John M.; O’Riordan, Eugene; Shishkin, G. I.

doi:10.1201/9781482285727

Cited by 614 publications

(546 citation statements)

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“…Finally we recall that the pseudo bubble functions B * i (i = 1, 2) are approximations to B i on the sub-grid specified above, through (12) and they are used in place of B i to represent u B in (11). The approximate representation of u B by bubble functions B * i (i = 1, 2) is eventually used to solve (7) for its linear part.…”

Section: Remarkmentioning

confidence: 99%

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Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

Sendur

Neslitürk

2011

Calcolo

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It is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the SUPG-type for the convection-diffusion-reaction problems. In particular, the residual-free bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless the configuration is simple. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. Here we propose a stabilization technique based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to L 1 norm. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. This fact is also confirmed by numerical experiments. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one in Brezzi et al. (Math. Models Methods Appl. Sci. 13:445-461, 2003).

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Section: Remarkmentioning

confidence: 99%

“…Yet another way of stabilizing the Galerkin method is to stabilize by means of a suitable refinement around the layer so that, the stabilization is actually not needed anymore, like in the Shishkin meshes [11]. The drawback of these methodologies resides in that they require a priori knowledge of the layer locations.…”

Section: Introductionmentioning

confidence: 99%

Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

Sendur

Neslitürk

2011

Calcolo

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“…They are very fine inside the boundary layer and coarse outside. Moreover, in 1990s the Russian mathematician Shishkin showed that one could use a simpler piecewise uniform mesh to obtain reasonable approximations [14,36]. This idea has been propagated throughout the 1990s by a group of Irish mathematicians: Miller, O'Riordan and Farrell [29].…”

Section: Introductionmentioning

confidence: 99%

“…Another major approach to obtain reasonable approximations for the CDR problem is the finite element method (FEM) [2,14,21]. The most successful classes of FEMs for treating convection-dominated problems are achieved by the stabilized formulations [16,18,19,22,24,37].…”

Section: Introductionmentioning

confidence: 99%

Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid

Kaya

Sendur

2015

Journal of Computational Physics

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A numerical scheme for the convection-diffusion-reaction (CDR) problems is studied herein. We propose a finite difference method on a special grid for solving CDR problems particularly designed to treat the most interesting case of small diffusion. We use the subgrid nodes in the Link-cutting bubble (LCB) strategy [5] to construct a numerical algorithm that can easily be extended to the higher dimensions. The method adapts very well to all regimes with continuous transitions from one regime to another. We also compare the performance of the present method with the Streamline-upwind PetrovGalerkin (SUPG) and the Residual-Free Bubbles (RFB) methods on several benchmark problems. The numerical experiments confirm the good performance of the proposed method.

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“…However, conditions imposed on the problem data (precisely, on their smoothness and the order of compatibility conditions on nonsmooth parts of the boundary), which ensures that the solution is sufficiently smooth, are restrictive and, as a rule, are substantially overstated so as to make it difficult to use these methods in practice. For example, in [13] (see also [3,8,10]) for a reaction-diffusion problem a sufficiently high level of smoothness of the coefficients and source terms (from the class C l (G), l > 6) and the boundary functions (from the class C l (S j ) ∩ C(S), l > 6, where S j are the sides forming the boundary S of the set G) was required in order to comply with sufficient conditions of ε-uniform convergence. If the fulfilment of the compatibility conditions at the edges (in three dimensions) or corners (in two dimensions) from S is not assumed, except continuity of the boundary function, then the order of ε-uniform convergence for the scheme on piecewise uniform meshes which was studied in [13] .…”

Section: Introductionmentioning

confidence: 99%

Approximation of Singularly Perturbed Parabolic Reaction-Diffusion Equations with Nonsmooth Data

Шишкин

2001

Comput. Methods Appl. Math.

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-In this paper we consider the Dirichlet problem on a rectangle for singularly perturbed parabolic equations of reaction-diffusion type. The reduced (for ε = 0) equation is an ordinary differential equation with respect to the time variable; the singular perturbation parameter ε may take arbitrary values from the half-interval (0,1]. Assume that sufficiently weak conditions are imposed upon the coefficients and the right-hand side of the equation, and also the boundary function. More precisely, the data satisfy the Hölder continuity condition with a small exponent α and α/2 with respect to the space and time variables. To solve the problem, we use the known ε-uniform numerical method (i.e., a standard finite difference operator on piecewiseuniform fitted meshes over the axes x 1 and x 2 ) which was developed previously for problems with sufficiently smooth and compatible data. It is shown that the numerical solution converges ε-uniformly at the rate ofhere the values of N and N 0 define the number of nodes in the space (with respect to each variable) and time meshes. We discuss also the behavior of local accuracy of the scheme in the case where the data of the boundary-value problem are smoother on a part of the domain of definition.2000 Mathematics Subject Classification: 65N06; 65N22; 35B25.Keywords: singular perturbation, finite difference scheme, fitted mesh, ε-uniform convergence. IntroductionAs is known, depending on the data of a boundary-value problem, in particular, on their smoothness, the solutions of singularly perturbed problems can be sufficiently smooth for each fixed value of the perturbation parameter ε multiplying the highest derivatives. However, the derivatives of solutions grow without bounds (in boundary and transient layers) as the parameter ε tends to zero. This is the reason why numerical methods developed for regular problems (see, e.g. [7,9]) yield errors which depend on an inverse power of the perturbation parameter ε; such errors become large for small ε which is clearly unsatisfactory. Approximation of singularly perturbed equations with non-smooth data 299For certain problems having sufficiently smooth solutions, ε-uniform numerical methods which converge irrespective of ε have been developed and thoroughly studied (see, e.g., [1,10] and the references therein) with the convergence analysis given in the maximum norm. However, conditions imposed on the problem data (precisely, on their smoothness and the order of compatibility conditions on nonsmooth parts of the boundary), which ensures that the solution is sufficiently smooth, are restrictive and, as a rule, are substantially overstated so as to make it difficult to use these methods in practice. For example, in [13] (see also [3,8,10]) for a reaction-diffusion problem a sufficiently high level of smoothness of the coefficients and source terms (from the class C l (G), l > 6) and the boundary functions (from the class C l (S j ) ∩ C(S), l > 6, where S j are the sides forming the boundary S of the set G) was required in o...

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Robust Computational Techniques for Boundary Layers

Cited by 614 publications

References 0 publications

Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid

Approximation of Singularly Perturbed Parabolic Reaction-Diffusion Equations with Nonsmooth Data

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