共查询到20条相似文献,搜索用时 21 毫秒
1.
《Journal of Computational and Applied Mathematics》2002,145(1):151-166
A Dirichlet problem for a system of two coupled singularly perturbed reaction–diffusion ordinary differential equations is examined. A numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed. Numerical results are presented, which illustrate the theoretical results. 相似文献
2.
In the present paper we analyse a finite element method for a singularly perturbed convection–diffusion problem with exponential
boundary layers. Using a mortaring technique we combine an anisotropic triangulation of the layer region (into rectangles)
with a shape regular one of the remainder of the domain. This results in a possibly non-matching (and hybrid), but layer adapted
mesh of Shishkin type. We study the error of the method allowing different asymptotic behaviour of the triangulations and
prove uniform convergence and a supercloseness property of the method. Numerical results supporting our analysis are presented. 相似文献
3.
Aastha Gupta Aditya Kaushik Manju Sharma 《Numerical Methods for Partial Differential Equations》2023,39(2):1220-1250
We propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin-boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature-based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first-order in time and second-order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis. 相似文献
4.
Pankaj Mishra Kapil K. Sharma Amiya K. Pani Graeme Fairweather 《Numerical Methods for Partial Differential Equations》2020,36(3):495-523
Quasi-optimal error estimates are derived for the continuous-time orthogonal spline collocation (OSC) method and also two discrete-time OSC methods for approximating the solution of 1D parabolic singularly perturbed reaction–diffusion problems. OSC with C1 splines of degree r ≥ 3 on a Shishkin mesh is employed for the spatial discretization while the Crank–Nicolson method and the BDF2 scheme are considered for the time-stepping. The results of numerical experiments validate the theoretical analysis and also exhibit additional quasi-optimal results, in particular, superconvergence phenomena. 相似文献
5.
We consider a singularly perturbed elliptic problem with two small parameters posed on the unit square. Based on a decomposition of the solution, we prove uniform convergence of a finite element method in an energy norm. The method uses piecewise bilinear functions on a layer-adapted Shishkin mesh. Numerical results confirm our theoretical analysis. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
6.
A procedure for the construction of robust, upper bounds for the error in the finite element approximation of singularly perturbed
reaction–diffusion problems was presented in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) which entailed the solution of an infinite dimensional local boundary value problem. It is not possible to solve this problem
exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a
subsequent article. We view the present work as fulfilling that promise and as completing the investigation begun in Ainsworth
and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) by removing the obligation to solve a local problem exactly. The resulting new estimator is indeed fully computable and
the first to provide fully computable, robust upper bounds in the setting of singularly perturbed problems discretised by
the finite element method. 相似文献
7.
Ahmed Al-Taweel Saqib Hussain Xiaoshen Wang Brian Jones 《Numerical Methods for Partial Differential Equations》2020,36(2):213-227
This paper investigates the lowest-order weak Galerkin finite element (WGFE) method for solving reaction–diffusion equations with singular perturbations in two and three space dimensions. The system of linear equations for the new scheme is positive definite, and one might readily get the well-posedness of the system. Our numerical experiments confirmed our error analysis that our WGFE method of the lowest order could deliver numerical approximations of the order O(h1/2) and O(h) in H1 and L2 norms, respectively. 相似文献
8.
We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for convergence in the L∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp estimates for the discrete Green function associated with the discretization.
AMS subject classification 65L10, 65L12, 65L15 相似文献
9.
《Journal of Computational and Applied Mathematics》2004,166(1):321-341
Reaction–diffusion problems have been used to describe pattern formation in developmental biology and material science. I study the existence and stability of a singularly perturbed reaction–diffusion problem with inhomogeneous environment in one-dimensional space domain and also high-dimensional space domain. 相似文献
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We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection–reaction–diffusion equation. Equations of this type arise in many contexts, such as for example the modeling of contaminant transport in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized using a recently developed hybrid mimetic mixed framework. We construct a family of discretizations for the convection term, which uses the hybrid interface unknowns. We consider a wide range of unstructured possibly nonmatching polyhedral meshes in arbitrary space dimension. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Péclet number. We obtain a convergence result based upon a priori estimates and the Fréchet–Kolmogorov compactness theorem. We implement the scheme both in two and three space dimensions and compare the numerical results obtained with the upwind and the centered discretizations of the convection term numerically. 相似文献
13.
In this article, a singularly perturbed convection–diffusion equation is solved by a linear finite element method on a Shishkin mesh. By means of an analysis exploiting symmetries in the convective term of the bilinear form, a new superconvergence rate, which improves the existing result, is obtained. 相似文献
14.
Justin B. Munyakazi 《Journal of Difference Equations and Applications》2013,19(5):799-813
We consider a class of singularly perturbed elliptic problems posed on a unit square. These problems are solved by using fitted mesh methods by many researchers but no attempts are made to solve them using fitted operator methods, except our recent work on reaction–diffusion problems [J.B. Munyakazi and K.C. Patidar, Higher order numerical methods for singularly perturbed elliptic problems, Neural Parallel Sci. Comput. 18(1) (2010), pp. 75–88]. In this paper, we design two fitted operator finite difference methods (FOFDMs) for singularly perturbed convection–diffusion problems which possess solutions with exponential and parabolic boundary layers, respectively. We observe that both of these FOFDMs are ?-uniformly convergent. This fact contradicts the claim about singularly perturbed convection–diffusion problems [Miller et al. Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996] that ‘when parabolic boundary layers are present, …, it is not possible to design an ?-uniform FOFDM if the mesh is restricted to being a uniform mesh’. We confirm our theoretical findings through computational investigations and also found that we obtain better results than those of Linß and Stynes [Appl. Numer. Math. 31 (1999), pp. 255–270]. 相似文献
15.
A semilinear reaction–diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive
parameter e2{\varepsilon^2} , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers
is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult. To address this
difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on
suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub-
and super-solutions. Convergence results are deduced that depend on the relative sizes of e{\varepsilon} and N, where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues
in using Newton’s method to compute a discrete solution are discussed. 相似文献
16.
This paper analyzes the implicit upwind finite difference scheme on Shishkin-type meshes (including the classical piecewise-uniform
Shishkin mesh and the Bakhalov-Shishkin mesh) for a class of singularly perturbed parabolic convection-diffusion problems
exhibiting strong interior layers. Suitable conditions on the mesh-generating functions are derived and are found to be sufficient
for the convergence of the method, uniformly with respect to the perturbation parameter. Utilizing these conditions, it is
shown that the method converges uniformly in the discrete supremum norm with an optimal error bound. Numerical results are
presented to validate the theoretical results. 相似文献
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In this paper, we propose a least-squares mixed element procedure for a reaction–diffusion problem based on the first-order system. By selecting the least-squares functional properly, the resulting procedure can be split into two independent symmetric positive definite schemes, one of which is for the unknown variable and the other of which is for the unknown flux variable, which lead to the optimal order H1(Ω) and L2(Ω) norm error estimates for the primal unknown and optimal H(div;Ω) norm error estimate for the unknown flux. Finally, we give some numerical examples. 相似文献
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A system of two coupled singularly perturbed convection–diffusion ordinary differential equations is examined. The diffusion
term in each equation is multiplied by a small parameter, and the equations are coupled through their convective terms. The
problem does not satisfy a conventional maximum principle. Its solution is decomposed into regular and layer components. Bounds
on the derivatives of these components are established that show explicitly their dependence on the small parameter. A numerical
method consisting of simple upwinding and an appropriate piecewise-uniform Shishkin mesh is shown to generate numerical approximations
that are essentially first order convergent, uniformly in the small parameter, to the true solution in the discrete maximum
norm.
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