A Comparison of Uniformly Convergent Difference Schemes for Two-Dimensional Convection—Diffusion Problems

Hegarty, Alan F.; O’Riordan, Eugene; Stynes, Martin

doi:10.1006/jcph.1993.1050

Cited by 42 publications

(29 citation statements)

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“…(as in [19]). It is clear from these results that, on both grids, the bound on the interpolation error (16) is sharp, that is, the error decays in practice like N −2 as predicted.…”

Section: Mesh Convergencementioning

confidence: 98%

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Ramage¹,

Newton²

2008

Molecular Crystals and Liquid Crystals

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The Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output. AbstractThis paper illustrates the use of moving mesh methods for solving partial differential equation (PDE) problems in Q-tensor theory of liquid crystals. We present the results of an initial study using a simple one-dimensional test problem which illustrates the feasibility of applying adaptive grid techniques in such situations. We describe how the grids are computed using an equidistribution principle, and investigate the comparative accuracy of adaptive and uniform grid strategies, both theoretically and via numerical examples.

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“…(as in [19]). It is clear from these results that, on both grids, the bound on the interpolation error (16) is sharp, that is, the error decays in practice like N −2 as predicted.…”

Section: Mesh Convergencementioning

confidence: 98%

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Ramage¹,

Newton²

2008

Molecular Crystals and Liquid Crystals

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“…Since the layers are typically exponential, another approach is to modify the differencing scheme in such a way that an exponential function is captured exactly rather than a polynomial one. This idea of "exponential fitting" is very old, dating back to 1950s [Allen and Sourthwell (1955)], and there have been many variants on this theme ; Hegarty et al (1993); Lube (1992); Sacco and Stynes (1998); Roos et al (1996)]. In the finite element setting, the finite element space can be modified with basis functions that have the exponential behavior.…”

Section: Convection-diffusion Equationmentioning

confidence: 99%

“…Among the most effective methods are the finite element methods with basis functions that contain the exponential behavior resembling the boundary layer of the solution [Roos et al (1991)]. One can, for example, obtain the trial functions by solving the homogeneous equation modified by making the coefficient constant [Hegarty et al (1993)]. In one dimension,…”

Section: Green's Function Approachmentioning

confidence: 99%

“…This works well for a small class of problems whose behavior is essentially one-dimensional, e.g., when ; Hegarty et al (1993)]. Other methods that work well in genuinely two-dimensional problems impose stringent conditions on the coefficients or require that much information is given about the layer structure in advance [Miller et al (1996)].…”

Section: Introductionmentioning

confidence: 99%

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Multiscale Numerical Methods for Singularly Perturbed Convection-Diffusion Equations

Park

Hou

2004

Int. J. Comput. Methods

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We present an efficient and robust approach in the finite element framework for numerical solutions that exhibit multiscale behavior, with applications to singularly perturbed convection-diffusion problems. The first type of equation we study is the convectiondominated convection-diffusion equation, with periodic or random coefficients; the second type of equation is an elliptic equation with singularities due to discontinuous coefficients and non-smooth boundaries. In both cases, standard methods for purely hyperbolic or elliptic problems perform poorly due to sharp boundary and internal layers in the solution.We propose a framework in which the finite element basis functions are designed to capture the local small-scale behavior correctly. When the structure of the layers can be determined locally, we apply the multiscale finite element method, in which we solve the corresponding homogeneous equation on each element to capture the small scale features of the differential operator. We demonstrate the effectiveness of this method by computing the enhanced diffusivity scaling for a passive scalar in the cellular flow. We also carry out the asymptotic error analysis for its convergence rate and perform numerical experiments for verification. For a random flow with nonlocal layer structure, we use a variational principle to gain additional information in our attempt to design asymptotic basis functions. We also apply the same framework for elliptic equations with discontinuous coefficients or non-smooth boundaries. In that case, we construct local basis function near singularities using infinite element method in order to resolve extreme singularity. Numerical results on problems with various singularities confirm the efficiency and accuracy of this approach.

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“…The SG method is based on the exact solution of locally constant flux equations and can be shown to be equivalent to the exponentially fitted Il'in-Allen-Southwell finite difference method [7]. Several other exponentially fitted schemes have been devised to generalize the one-dimensional SG method to two dimensions: finite element methods that use approximate L-splines (i.e., trial functions that lie locally in the null space of an approximation of L) [8][9][10][11][12][13], mixed finite element and finite volume methods [14][15][16][17], finite volume methods [18,19] and exponentially fitted differences schemes [20] are some attempts made in the recent literature. Finite element methods designed to enjoy a discrete maximum principle have also been proposed in [21][22][23].…”

Section: Introductionmentioning

confidence: 99%

A nonconforming exponentially fitted finite element method for two-dimensional drift-diffusion models in semiconductors

Sacco

Gatti

Gotusso

1999

Numer. Methods Partial Differential Eq.

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A new nonconforming exponentially fitted finite element for a Galerkin approximation of convectiondiffusion equations with a dominating advective term is considered. The attention is here focused on the drift-diffusion current continuity equations in semiconductor device modeling. The scheme extends to the two-dimensional case, the well known Scharfetter-Gummel method, by imposing a divergence-free current over each element of the triangulation. Convergence of the method in the energy norm is proved and some numerical results are included.

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A Comparison of Uniformly Convergent Difference Schemes for Two-Dimensional Convection—Diffusion Problems

Cited by 42 publications

References 0 publications

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Multiscale Numerical Methods for Singularly Perturbed Convection-Diffusion Equations

A nonconforming exponentially fitted finite element method for two-dimensional drift-diffusion models in semiconductors

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