Parameter-uniform  finite difference schemes    for singularly perturbed parabolic   diffusion-convection-reaction problems

O’Riordan, Eugene; Pickett, M. L.; Шишкин, Г. И.

doi:10.1090/s0025-5718-06-01846-1

Cited by 70 publications

(45 citation statements)

References 13 publications

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“…In such cases, the accuracy of the solution also decreases. At high speeds, numerical methods based on the separation of the flow direction and boundary layers should be used [12,13]. Our method is designed to solve problems with a relatively slow movements of the complex structure when the implementation of the scheme with the separation of the flow direction and boundary layers is difficult.…”

Section: Heat Transfer In the Microchipmentioning

confidence: 99%

Energy Method for Mathematical Modeling of Heat Transfer in 2-D Flow

Денисенко¹,

Денисенко²

2014

J. Sib. Fed. Univ., Math. phys.

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A multigrid finite element method for 2-D convection-heat transfer problem is proposed. The method is based on minimization of the energy functional. Effectiveness of the method is demonstrated by calculation of temperature distribution in a microchip that separates living cells from fluid stream by electrophoresis.Keywords: mathematical modeling, energy method, elliptical equation, nonsymmetric operator, electrophoresisThe problem of heat transfer in a moving medium and similar convection-diffusion problem arise in various fields of research, for example, in mathematical modeling of the transport of pollutants and energy in the atmosphere or in mathematical modeling of devices created for biological research.Heat transfer is described by second order elliptic equation with proportional to velocity first order term. Then operators of boundary value problems are asymmetric operators. Because of this, the principle of minimum of a quadratic energy functional is not formulated. For problems with symmetric operators this principle allows one to use the most effective methods for the approximate and numerical solutions [10]. For problems with asymmetric operator the principles of nonequilibrium thermodynamics of minimal entropy production are not formulated, etc. [11]. Some problems with asymmetric operators are reformulated as problems with symmetric positive definite operators, and for them the energy principles are proved in [2,4]. Three-dimensional problems are discussed in [3].The aim of this work is to create a multigrid finite element method for solving two-dimensional heat transfer problem in a moving medium and to demonstrate effectiveness of the method.As an example temperature distribution in the microchip, that separates living cells from a fluid stream by means of electrophoresis is calculated.

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Section: Heat Transfer In the Microchipmentioning

confidence: 99%

Energy Method for Mathematical Modeling of Heat Transfer in 2-D Flow

Денисенко¹,

Денисенко²

2014

J. Sib. Fed. Univ., Math. phys.

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“…An appropriate layer-adapted mesh can be constructed based on the mesh given in [16] and a suitable fitted operator would be R(μā).…”

Section: Remarkmentioning

confidence: 99%

A parameter robust Petrov–Galerkin scheme for advection–diffusion–reaction equations

Falco

O’Riordan

2010

Numer Algor

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In this paper, a singularly perturbed convection diffusion boundary value problem, with discontinuous diffusion coefficient is examined. In addition to the presence of boundary layers, strong and weak interior layers can also be present due to the discontinuities in the diffusion coefficient. A priori layer adapted piecewise uniform meshes are used to resolve any layers present in the solution. Using a Petrov-Galerkin finite element formulation, a fitted finite difference operator is shown to produce numerical approximations on this fitted mesh, which are uniformly second order (up to logarithmic terms) globally convergent in the pointwise maximum norm.

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“…Parameter-uniform numerical methods have been analysed for the onedimensional two-parameter problem in [12][13][14]. In this paper, we extend these ideas to a two parameter problem in two space dimensions.…”

Section: Introductionmentioning

confidence: 99%

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

O’Riordan

Pickett

2010

Adv Comput Math

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In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established.

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Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems

Cited by 70 publications

References 13 publications

Energy Method for Mathematical Modeling of Heat Transfer in 2-D Flow

Energy Method for Mathematical Modeling of Heat Transfer in 2-D Flow

A parameter robust Petrov–Galerkin scheme for advection–diffusion–reaction equations

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

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