A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem

Kopteva, Natalia; Stynes, Martin

doi:10.1137/s003614290138471x

Cited by 115 publications

(88 citation statements)

References 21 publications

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“…Many people use the arc-length as the monitor function, namely M = 1 + |u | 2 or its discrete analogue [12,13,24,29]. The first order uniform convergence on the equidistributed grid is obtained even for the fully adapted algorithm [13,24].…”

Section: Introductionmentioning

confidence: 99%

An optimal streamline diffusion finite element method for a singularly perturbed problem

Chen¹,

Xu²

2005

Recent Advances in Adaptive Computation

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Abstract. The stability and accuracy of a streamline diffusion finite element method (SDFEM) on arbitrary grids applied to a linear 1-d singularly perturbed problem are studied in this paper. With a special choice of the stabilization quadratic bubble function, the SDFEM is shown to have an optimal second order in the sense thatwhere u h is the SDFEM approximation of the exact solution u and V h is the linear finite element space. With the quasi-optimal interpolation error estimate, quasi-optimal convergence results for the SDFEM are obtained. As a consequence, an open question about the optimal choice of the monitor function for a second order scheme in the moving mesh method is answered.

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Section: Introductionmentioning

confidence: 99%

An optimal streamline diffusion finite element method for a singularly perturbed problem

Chen¹,

Xu²

2005

Recent Advances in Adaptive Computation

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“…In fact, as lim | − |→0 ( ) [ , ] /| − | = | ( )|, we can observe an equidistribution of the solution gradient | ( )| in the range-discrete grid. This is an obvious adaptive criterion widely used in many adaptive mesh methods [19,28,29]. While complex adaptive operations have to be involved in these methods, it is done by the range-discrete mesh simply via a equidistance discretization in value domain.…”

Section: Range-discrete Meshmentioning

confidence: 99%

A Self‐Adaptive Numerical Method to Solve Convection‐Dominated Diffusion Problems

Cao

Liu

et al. 2017

Mathematical Problems in Engineering

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Convection-dominated diffusion problems usually develop multiscaled solutions and adaptive mesh is popular to approach high resolution numerical solutions. Most adaptive mesh methods involve complex adaptive operations that not only increase algorithmic complexity but also may introduce numerical dissipation. Hence, it is motivated in this paper to develop an adaptive mesh method which is free from complex adaptive operations. The method is developed based on a range-discrete mesh, which is uniformly distributed in the value domain and has a desirable property of self-adaptivity in the spatial domain. To solve the time-dependent problem, movement of mesh points is tracked according to the governing equation, while their values are fixed. Adaptivity of the mesh points is automatically achieved during the course of solving the discretized equation. Moreover, a singular point resulting from a nonlinear diffusive term can be maintained by treating it as a special boundary condition. Serval numerical tests are performed. Residual errors are found to be independent of the magnitude of diffusive term. The proposed method can serve as a fast and accuracy tool for assessment of propagation of steep fronts in various flow problems.

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“…The moving mesh method [6] will be used to solve . The drawback of this strategy is that, with the introduction of the mesh equations which govern mesh movement, the system becomes nonlinear for any linear problem; hence very little theoretical analysis [1,7,18,19] has been carried out to explain the convergence behaviour of the method. The following assumptions will be made: for all ( , ) ∈ Ω, ‖ ( )/ ‖ ≤ , for some constant and at = T, ‖ ( , )/ ‖ ∼ ( ) and ‖ ( )/ ‖ ∼ ( ).…”

Section: Abstract and Applied Analysismentioning

confidence: 99%

“…The use of adapted meshes [1][2][3] in the numerical solution of differential equations has become a popular technique for improving existing approximation schemes. When considering an adaptive mesh algorithm for the solution of time dependent differential equations [4][5][6], the techniques which underpin the grid movement are often found in the literature [4,6,7] for the generation of adapted grids for the numerical solution of steady problems.…”

Section: Introductionmentioning

confidence: 99%

A Moving Mesh Method for Singularly Perturbed Problems

Sikwila¹,

Shateyi

2013

Abstract and Applied Analysis

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A singularly perturbed time dependent convection diffusion problem is solved on a rectangular domain, using the moving mesh method which uses the equidistribution principle. The problem has a boundary at the steady state. It is shown that the numerical approximations generated by the moving mesh method converge uniformly with respect to the singular perturbation parameter. Theoretical results are obtained which are verified using numerical results.

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A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem

Cited by 115 publications

References 21 publications

An optimal streamline diffusion finite element method for a singularly perturbed problem

An optimal streamline diffusion finite element method for a singularly perturbed problem

A Self‐Adaptive Numerical Method to Solve Convection‐Dominated Diffusion Problems

A Moving Mesh Method for Singularly Perturbed Problems

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