Perturbed Reaction-diffusion Problem Research Articles

A simple approach to reliable and robust a posteriori error estimation for singularly perturbed problems

A simple flux reconstruction for finite element solutions of reaction–diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction–diffusion problem. The flux reconstruction is based on simple, independent post-processing operations over patches of elements in conjunction with standard Raviart–Thomas vector fields. We prove that the corresponding local error indicators are locally efficient and robust (up to a nonstandard oscillation term) with respect to any mesh size and any size of the reaction coefficient, including the singularly perturbed limit.

Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems

Abstract We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.

Numerical solutions of a system of singularly perturbed reaction-diffusion problems

This paper addresses the numerical approximation of solutions to a coupled system of singularly perturbed reaction-diffusion equations. The components of the solution exhibit overlapping boundary and interior layers. Sinc procedure can control the oscillations in computed solutions at boundary layer regions naturally because the distribution of Sinc points is denser at near the boundaries. Also the obtained results show that the proposed method is applicable even for small perturbation parameter as ? = 2-30. The convergence analysis of proposed technique is discussed, it is shown that the approximate solutions converge to the exact solutions at an exponential rate. Numerical experiments are carried out to demonstrate the accuracy and efficiency of the method.

Finite element approximation of reaction–diffusion problems using an exponentially graded mesh

We present the analysis of an h version Finite Element Method for the approximation of the solution to singularly perturbed reaction–diffusion problems posed in smooth domains Ω⊂R2. The method uses piecewise polynomials of degree p in each variable, defined on an exponentially graded mesh, optimally constructed for the approximation of exponential layers. We establish robust, optimal convergence rates in a variety of norms and illustrate our theoretical findings through numerical computations.

A posteriori maximum-norm error bounds for the biquadratic spline collocation method applied to reaction-diffusion problems

Collocation with biquadratic $$C^1$$ -splines for a singularly perturbed reaction-diffusion problem in two dimensions is studied. A second-order supremum norm a posteriori error bound is derived for the collocation method on an arbitrary mesh. Numerical results are presented that support our theoretical estimate.

Parameter-Robust Numerical Scheme for Time-Dependent Singularly Perturbed Reaction–Diffusion Problem with Large Delay

ABSTRACTThis work presents the development of numerical scheme for second-order time-dependent singularly perturbed reaction-diffusion problem with large delay in the undifferentiated term. These types of problems arise frequently in many areas of science and engineering that take into consideration the effect of present situation as well as the past history of the physical system. As the characteristics of the reduced problem (𝜀 = 0) corresponding to the original singularly perturbed problem considered here are parallel to the boundary of the domain this implies, parabolic layers exhibit in the solution. In this paper, we initiate the study of parabolic layers together with interior layers in the solution of singularly perturbed parabolic partial differential-difference equations due to propagation of singularity. Proposed numerical scheme comprised of finite difference scheme and piecewise uniform Shishkin mesh. The method is shown to be accurate of order , where M and N are the number of mesh elements in time and spatial direction, respectively. Proposed numerical scheme is proved to be parameter uniform convergent in the maximum norm. Numerical experiments have been performed to show the existence of interior layer due to large state-dependent delay argument in the reaction term and to confirm the predicted theory.

A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer

In this paper, we consider a singularly perturbed reaction-diffusion problem with a discontinuous source term. Boundary and interior layers appear in the solution. The problem is discretized by using a hybrid finite difference scheme on a Shishkin-type mesh. A nonequidistant generalization of the Numerov scheme is used on the Shishkin-type mesh except for the point of discontinuity, whereas a second-order difference scheme with an additional refined mesh is used for the point of discontinuity. Although the difference scheme does not satisfy the discrete maximum principle, the maximum norm stability of the scheme is established. The maximum error in the mesh points is shown to be uniformly bounded by ( N^{-1}ln N ) ^{4} with a constant independent of the perturbation parameter. Numerical results supporting the theory are presented.

Parameter-uniform numerical methods for general nonlinear singularly perturbed reaction diffusion problems having a stable reduced solution

A general nonlinear singularly perturbed reaction diffusion differential equation with solutions exhibiting boundary layers is analysed in this paper. The problem is considered as having a stable (attractive) reduced solution that satisfies any one of a comprehensive set of conditions for stable reduced solutions of reaction diffusion problems. A numerical method is presented consisting of a finite difference scheme to be solved over a Shishkin mesh. It is shown that suitable transition points for the Shishkin mesh and the error of the numerical method depend on which stability condition the reduced solution satisfies. Moreover, we show that the error may be affected adversely depending on the stability condition satisfied. Numerical experiments are presented to demonstrate the convergence rate established.

Higher-order accurate schemes for a constant coefficient singularly perturbed reaction-diffusion problem of boundary layer type

Higher-order accurate schemes for a constant coefficient singularly perturbed reaction-diffusion problem of boundary layer type

Higher-order accurate schemes for a variable coefficient singularly perturbed reaction-diffusion problem of boundary layer type

Higher-order accurate schemes for a variable coefficient singularly perturbed reaction-diffusion problem of boundary layer type

A variational-perturbation method for solving the time-dependent singularly perturbed reaction-diffusion problems

In this paper, we combine the variational iteration method and perturbation theory to solve a time-dependent singularly perturbed reaction-diffusion problem. The problem is considered in the boundary layers and outer region. In the boundary layers, the problem is transformed by the variable substitution, and then the variational iteration method is employed to solve the transformed equation. In the outer region, we use the perturbation theory to obtain the approximation equation and the approximation solution. The final numerical experiments show that this method is accurate.

Higher-order accurate schemes for a constant coefficient singularly perturbed reaction-diffusion problem of highly oscillatory type

Higher-order accurate schemes for a constant coefficient singularly perturbed reaction-diffusion problem of highly oscillatory type

Higher-order accurate schemes for a variable coefficient singularly perturbed reaction-diffusion problem of highly oscillatory type

Higher-order accurate schemes for a variable coefficient singularly perturbed reaction-diffusion problem of highly oscillatory type

An adapted Petrov–Galerkin multi-scale finite element method for singularly perturbed reaction–diffusion problems

In this paper, we propose the use of an adapted Petrov–Galerkin (PG) multi-scale finite element method for solving the singularly perturbed problem. The multi-scale basis functions that form the function space are constructed from both homogeneous and nonhomogeneous localized problems, which provide more flexibility. These PG multi-scale basis functions are shown to capture the originally perturbed information for the reaction–diffusion model, and reduce the boundary layer errors on graded (non-uniform) coarse meshes. We present the numerical experiment in order to demonstrate that our method acquires stable and convergent results in the , and energy norms. Due to the independent construction of the multi-scale bases, and the demonstrated accuracy by removing the resonance effect, the adapted PG multi-scale method is shown to be a suitable method for solving the singular perturbation problem.

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.

An analysis of overlapping domain decomposition methods for singularly perturbed reaction–diffusion problems

In this paper, we present an analysis of overlapping domain decomposition methods for singularly perturbed reaction–diffusion problems. For this purpose, we consider a model problem for coupled system of singularly perturbed reaction–diffusion equations with distinct small positive parameters, exhibiting overlapping boundary layers at both ends of the domain. A discrete overlapping Schwarz domain decomposition method is considered to solve the model problem numerically. The analysis is based on defining some auxiliary problems that allow to prove the uniform convergence of the method in two steps, splitting the discretization error and the iteration error. The numerical approximations generated from the method are shown to be almost second order uniformly convergent. The presented idea has a general significance and can be used for the analysis of overlapping domain decomposition methods for other classes of singularly perturbed reaction–diffusion problems.

A multiscale sparse grid finite element method for a two-dimensional singularly perturbed reaction-diffusion problem

We consider the numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem posed on the unit square by a multiscale sparse grid finite element method. A Shishkin mesh which resolves the boundary and corner layers, and yields a parameter robust solution, is used. Our analysis shows that the method achieves essentially the same level of accuracy, in the energy norm, as the standard Galerkin finite element method with bilinear elements. However, only O(NlogN)$\mathcal {O}(N\log N)$ degrees of freedom are required, compared to O(N2)$\mathcal {O}(N^{2})$ for the corresponding Galerkin finite element method. This may be regarded as a generalisation of Liu et al. (IMA J. Numer. Anal. 29(4), 986---1007 2009) which used a two-scale method requiring O(N3/2)$\mathcal {O}(N^{3/2})$ degrees of freedom. Numerical results are provided that demonstrate the sharpness of the estimates and the efficiency of the method.

Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction–diffusion boundary-value problems

In this article, we study the problem of determining an appropriate grading of meshes for a system of coupled singularly perturbed reaction–diffusion problems having diffusion parameters with different magnitudes. The central difference scheme is used to discretize the problem on adaptively generated mesh where the mesh equation is derived using an equidistribution principle. An a priori monitor function is obtained from the error estimate. A suitable a posteriori analogue of this monitor function is also derived for the mesh construction which will lead to an optimal second-order parameter uniform convergence. We present the results of numerical experiments for linear and semilinear reaction–diffusion systems to support the effectiveness of our preferred monitor function obtained from theoretical analysis.

A Hybrid Difference Scheme for a Second-Order Singularly Perturbed Reaction-Diffusion Problem with Non-smooth Data

A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. In Miller et al. (J Appl Numer Math 35(4):323–337, 2000) the authors discussed problems that arises naturally in the context of models of simple semiconductor devices. Due to the discontinuity, interior layers appear in the solution. The problem is solved using a hybrid difference scheme on a Shishkin mesh. We prove that the method is second order convergent in the maximum norm, independently of the diffusion parameter. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

AbstractWe apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in theL2norm and first order convergence in the energy norm on graded meshes, which is independent of ɛ. In contrast with the conventional methods, our method is much more accurate and effective.