Perturbed Reaction-diffusion Problem Research Articles

Singularly Perturbed Reaction\u2013Diffusion Problems as First order systems

We consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and H_{{{,mathrm{{div}},}}}-conforming elements for the second component we provide a convergence analysis on layer adapted meshes and an optimal convergence order in a balanced norm that is comparable with a balanced H^2-norm for the second order formulation.

Error estimations in the balanced norm of finite element method on Bakhvalov–Shishkin triangular mesh for reaction–diffusion problems

A balanced norm, rather than the common energy norm, is introduced to reflect the behavior of layers more accurately in the finite element method for singularly perturbed reaction–diffusion problems. Convergence of optimal order in the balanced norm has been proved in the case of rectangular finite elements. However, for triangular finite elements Pk (k≥2), it is still open to prove this convergence result. With the help of the L2-stability of a weighted L2 projection, instead of the L∞-stability widely used in existing references, the geometric constraints on meshes are relaxed. As a result, the optimal order convergence in the balanced norm is proved in the case of Bakhvalov–Shishkin triangular meshes. Numerical experiments support theoretical results.

Convergence and Supercloseness in a Balanced Norm of Finite Element Methods on Bakhvalov-Type Meshes for Reaction-Diffusion Problems

In convergence analysis of finite element methods for singularly perturbed reaction–diffusion problems, balanced norms have been successfully introduced to replace standard energy norms so that layers can be captured. In this article, we focus on the convergence analysis in a balanced norm on Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel interpolation operator, which consists of a local $$L^2$$ projection operator and the Lagrange interpolation operator, is introduced for a convergence analysis of optimal order in the balanced norm. The analysis also depends on the stabilities of the $$L^2$$ projection and the characteristics of Bakhvalov-type meshes. Furthermore, we obtain a supercloseness result in the balanced norm, which appears in the literature for the first time. This result depends on another novel interpolant, which consists of the local $$L^2$$ projection operator, a vertices-edges-element operator and some corrections on the boundary.

A weighted and balanced FEM for singularly perturbed reaction-diffusion problems

A weighted and balanced FEM for singularly perturbed reaction-diffusion problems

A higher-order accurate difference approximation of singularly perturbed reaction-diffusion problem using grid equidistribution

This paper proposes a higher-order numerical approximation scheme to solve singularly perturbed reaction–diffusion boundary value problems. The proposed scheme is a combination of a fourth-order numerical difference method and classical central difference method applied on a non-equidistant grid. The non-equidistant grid is generated by equi-distributing a non-negative monitor function. The theoretical and empirical error analysis demonstrate that the proposed scheme has fourth-order uniform convergence with respect to the perturbation parameter.

Additive schemes based domain decomposition algorithm for solving singularly perturbed parabolic reaction-diffusion systems

In this paper, we examine a 1D parabolic coupled system of singularly perturbed reaction-diffusion problems in which perturbation parameters can be of distinct magnitude. To solve this system numerically we develop additive (or splitting) schemes based domain decomposition algorithm of Schwarz waveform relaxation type. On each subdomain we consider two additive schemes on a uniform mesh in time and the standard central difference scheme on a uniform mesh in space. We provide convergence analysis of the algorithm using some auxiliary problems and the algorithm is shown to be uniformly convergent. The additive schemes make the computation more efficient as they decouple the components of the approximate solution at each time level. Numerical results for two test problems are given in support of the theoretical convergence result and as well as to illustrate the efficiency of the additive schemes.

Finite difference scheme for singularly perturbed reaction diffusion problem of partial delay differential equation with nonlocal boundary condition

This paper investigates singularly perturbed parabolic partial differential equations with delay in space, and the right end plane is an integral boundary condition on a rectangular domain. A small parameter is multiplied in the higher order derivative, which gives boundary layers, and due to the delay term, one more layer occurs on the rectangle domain. A numerical method comprising the standard finite difference scheme on a rectangular piecewise uniform mesh (Shishkin mesh) of N_{r} times N_{t} elements condensing in the boundary layers is suggested, and it is proved to be parameter-uniform. Also, the order of convergence is proved to be almost two in space variable and almost one in time variable. Numerical examples are proposed to validate the theory.

The Novel Rational Spectral Collocation Method for a Singularly Perturbed Reaction-Diffusion Problem with Nonsmooth Data

A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. A novel rational spectral collocation combined with a singularity-separated method for this problem is presented. The solution is expressed as u = w + v , where w is the solution of corresponding auxiliary boundary value problem and v is a singular correction with direct expressions. The rational spectral collocation method combined with a sinh transformation is applied to solve the weakened singularly boundary value problem. According to the asymptotic analysis, the sinh transformation parameters can be determined by the width and position of the boundary layers. The parameters in the singular correction can be determined by the boundary conditions of the original problem. Numerical experiment supports theoretical results and shows that compared with previous research results, the novel method has the advantages of a high computational accuracy in singularly perturbed reaction-diffusion problems with nonsmooth data.

A modified graded mesh and higher order finite element method for singularly perturbed reaction–diffusion problems

This paper presents a modified graded mesh for singularly perturbed reaction–diffusion problems. The mesh we offer is generated recursively using Newton’s algorithm and some implicitly defined function. The problem is solved numerically using the finite element method based on polynomials of degree p≥1. We prove parameter uniform convergence of optimal order in ϵ-weighted energy norm. Test examples are taken, and we present a rigorous comparative analysis with other adaptive meshes. Moreover, we compare the proposed method with others found in the literature.

The finite volume element method on the Shishkin mesh for a singularly perturbed reaction–diffusion problem

In this paper, we construct a finite volume element method (FVEM) on the Shishkin mesh for solving a singularly perturbed reaction–diffusion problem. The stability of the method is established in energy norm. Further, we derive the optimal error estimate in energy norm, under the decomposition of solution. Some numerical experiments are provided to confirm our theoretical results.

Sinc-Galerkin method for solving system of singular perturbed reaction-diffusion problems

In this paper, the system of singularly Perturbed Reaction-Diffusion p roblems w hich a re commonly used in physics and chemistry branches of science, were investigated. Sinc-Galerkin Method was used to obtaining the solution of problems. Because of there is no article about Sinc-Galerkin Method related to singularly perturbed Reaction-Diffusion problems in liter- ature, the efficiency of the method was shown via this problem. There are important results that occurred after our research and application. Sinc Galerkin Method which was used in this paper as the main solution method gave better results according to parameter robust method and asymptotical initial value method. The figures and the tables show this competence and low errors.

Simple and robust equilibrated fluxa posterioriestimates for singularly perturbed reaction–diffusion problems

We consider energy norma posteriorierror analysis of conforming finite element approximations of singularly perturbed reaction–diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes orviacombination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtainedvialocal reaction–diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest.

Convergence of a finite element method on a Bakhvalov-type mesh for singularly perturbed reaction–diffusion equation

A finite element method is applied on a Bakhvalov-type mesh to solve a singularly perturbed reaction–diffusion problem whose solution exhibits boundary layers. A uniform convergence order of O(N−(k+1)+ε1/2N−k) is proved, where k is the order of piecewise polynomials in the finite element method, ε is the diffusion parameter and N is the number of partitions in each direction. Numerical experiments support this theoretical result.

A robust second order numerical method for a weakly coupled system of singularly perturbed reaction-diffusion problem with discontinuous source term

In this paper, a fitted mesh numerical method on Shishkin mesh is proposed to solve a weakly coupled system of two singularly perturbed reaction-diffusion equations containing equal diffusion parameters with discontinuous source terms. This method uses the standard centred finite difference scheme constructed on piecewise-uniform Shishkin mesh with the average of the source terms on either side of the point of discontinuity and then the problem is solved by an iterative procedure. An error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to confirm the theoretical results and compares well with the existing results.

A robust second order numerical method for a weakly coupled system of singularly perturbed reaction-diffusion problem with discontinuous source term

In this paper, a fitted mesh numerical method on Shishkin mesh is proposed to solve a weakly coupled system of two singularly perturbed reaction-diffusion equations containing equal diffusion parameters with discontinuous source terms. This method uses the standard centred finite difference scheme constructed on piecewise-uniform Shishkin mesh with the average of the source terms on either side of the point of discontinuity and then the problem is solved by an iterative procedure. An error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to confirm the theoretical results and compares well with the existing results.

A Dual Finite Element Method for a Singularly Perturbed Reaction-Diffusion Problem

We present a dual finite element method for a singularly perturbed reaction-diffusion problem. It can be considered a reduced version of the mixed finite element method for approximate solutions. The new method only approximates the dual variables without approximating the primary variable. An approximation for the primary variable is recovered through a simple local $L_2$ projection. Optimal error estimates for the primary and flux variables are obtained. Our method provides a competitive alternative to other existing numerical methods. For example, our approximate solution for the primary variable does not show a significant numerical oscillation, which is observed in the standard Galerkin methods, and we present a confirming numerical example.

Singularly perturbed reaction–diffusion problems with discontinuities in the initial and/or the boundary data

Numerical approximations to the solutions of three different problem classes of singularly perturbed parabolic reaction–diffusion problems, each with a discontinuity in the boundary-initial data, are generated. For each problem class, an analytical function associated with the discontinuity in the data, is identified. Parameter-uniform numerical approximations to the difference between the analytical function and the solution of the singularly perturbed problem are generated using piecewise-uniform Shishkin meshes. Numerical results are given to illustrate all the theoretical error bounds established in the paper.

High‐order discrete‐time orthogonal spline collocation methods for singularly perturbed 1D parabolic reaction–diffusion problems

AbstractQuasi‐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.

Maximum‐norm a posteriori error bounds for a collocation method applied to a singularly perturbed reaction–diffusion problem in three dimensions

AbstractCollocation with triquadratic C1‐splines for a singularly perturbed reaction–diffusion problem in three dimension is studied. A posteriori error bound in the maximum norm is derived for the collocation method on arbitrary tensor‐product meshes which is robust in the perturbation parameter. Numerical results are presented that support our theoretical estimate.

Uniformly convergent novel finite difference methods for singularly perturbed reaction–diffusion equations

AbstractIn this paper, we construct a kind of novel finite difference (NFD) method for solving singularly perturbed reaction–diffusion problems. Different from directly truncating the high‐order derivative terms of the Taylor's series in the traditional finite difference method, we rearrange the Taylor's expansion in a more elaborate way based on the original equation to develop the NFD scheme for 1D problems. It is proved that this approach not only can highly improve the calculation accuracy but also is uniformly convergent. Then, applying alternating direction implicit technique, the newly deduced schemes are extended to 2D equations, and the uniform error estimation based on Shishkin mesh is derived, too. Finally, numerical experiments are presented to verify the high computational accuracy and theoretical prediction.