Perturbed Reaction-diffusion Problem Research Articles

Error analysis of a weak Galerkin finite element method for singularly perturbed differential-difference equations

A weak Galerkin finite element method is applied to singularly perturbed delay reaction-diffusion problems. A robust uniform convergence has been proved both in the energy and balanced norms using higher-order piecewise discontinuous polynomials on Shishkin meshes. The error analysis for singularly perturbed reaction-diffusion problems with negative or positive shift in the balanced norm has appeared for the first time. The proposed method uses piecewise polynomials of order k ≥ 1 on interior of each element and piecewise constant polynomials on the end points of each element. By the Schur complement technique, the interior degrees of foredoom (DOF) can be eliminated from the discrete system resulting from the numerical scheme, and thus the degrees of freedom of the proposed method comparable with the classical finite element methods, and it is remarkably less than that of the discontinuous Galerkin method. Finally, we give various numerical experiments to verify the theoretical findings.

Fourth-order fitted mesh scheme for semilinear singularly perturbed reaction–diffusion problems

ObjectiveThe main purpose of this work is to present a fourth-order fitted mesh scheme for solving the semilinear singularly perturbed reaction–diffusion problem to produce more accurate solutions.ResultsQuasilinearization technique is used to linearize the semilinear term. The scheme is formulated with discretizing the solution domain piecewise uniformly and then replacing the differential equation by finite difference approximations. This gives the system of difference algebraic equations and is solved by the Thomas algorithm. Convergence analysis are investigated using solution bound and the truncation error bound. Numerical illustrations are investigated to support the theoretical results and the method’s applicability. The method produces a more accurate solution than some existing methods in the literature.

An improved pointwise error estimate of the LDG method for 1-d singularly perturbed reaction-diffusion problem

The local discontinuous Galerkin (LDG) method on a Shishkin mesh is studied for a one-dimensional singularly perturbed reaction-diffusion problem. Based on the discrete Green's function, we derive some improved pointwise error estimates in the regular and layer regions. For the LDG approximation to the solution itself, the convergence rate of the pointwise error in the regular region is sharp. For the LDG approximation to the derivative of the solution, the convergence rate of the pointwise error in the whole domain is optimal. We establish also optimal pointwise error estimates in the case that the regular component of the exact solution belongs to the finite element space. Numerical experiments are presented to validate the theoretical results.

Error Estimate in the Balanced Norm for the Singularly Perturbed Reaction–Diffusion Problems

Error Estimate in the Balanced Norm for the Singularly Perturbed Reaction–Diffusion Problems

Energy-norm and balanced-norm supercloseness error analysis of a finite volume method on Shishkin meshes for singularly perturbed reaction–diffusion problems

Energy-norm and balanced-norm supercloseness error analysis of a finite volume method on Shishkin meshes for singularly perturbed reaction–diffusion problems

Balanced and energy norm error bounds for a spatial FEM with Crank-Nicolson and BDF2 time discretisation applied to a singularly perturbed reaction-diffusion problem

Balanced and energy norm error bounds for a spatial FEM with Crank-Nicolson and BDF2 time discretisation applied to a singularly perturbed reaction-diffusion problem

A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift

ObjectiveThe paper is focused on developing and analyzing a uniformly convergent numerical scheme for a singularly perturbed reaction-diffusion problem with a negative shift. The solution of such problem exhibits strong boundary layers at the two ends of the domain due to the influence of the perturbation parameter, and the term with negative shift causes interior layer. The rapidly changing behavior of the solution in the layers brings significant difficulties in solving the problem analytically. We have treated the problem by proposing a numerical scheme using the implicit Euler method in the temporal direction and a fitted tension spline method in the spatial direction with uniform meshes.ResultStability and uniform error estimates are investigated for the developed numerical scheme. The theoretical finding is demonstrated by numerical examples. It is obtained that the developed numerical scheme is uniformly convergent of order one in time and order two in space.

Singularly perturbed reaction diffusion problem with large spatial delay via non-standard fitted operator method

In this article, a numerical method for singularly perturbed partial differential equations of the reaction-diffusion type with a large spatial delay is constructed. The method is developed using the Crank-Nicolson method for time derivative and a non-standard finite difference method for spatial derivative on a uniform mesh. Due to the presence of a small perturbation parameter, which multiplies the highest order derivative term, the solution exhibits twin boundary layers. The solution to the problem also shows an interior layer as a result of the delay in the spatial variable. To enhance the order of convergence, the Richardson extrapolation technique is applied. The convergence analysis of the proposed scheme is given. It is proved that the resulting scheme is uniformly convergent of fourth-order accurate in both space and time variables. Numerical experiments are conducted, and the results are in agreement with the theoretical findings. In addition, comparisons are performed, and the results show that the proposed scheme gives more accurate solutions and a higher rate of convergence than previous findings in the literature.

Supercloseness in a balanced norm of the NIPG method on Shishkin mesh for a reaction diffusion problem

For the error analysis of singularly perturbed reaction-diffusion problems, the balanced norm, which is stronger than the usual energy norm, is introduced to correctly reflect the behavior of the layer. In this paper, we study the convergence in a balanced norm for nonsymmetric interior penalty Galerkin (NIPG) method for the first time. For this purpose, a new interpolation is designed, which consists of a Gauß Lobatto interpolation in the layer, and a locally weighted L2 projection outside the layer. On that basis, by properly defining the penalty parameters at different nodes on a Shishkin mesh, we obtain the supercloseness of almost k+12 order, and prove the convergence of optimal order in a balanced norm. Here k is the degree of polynomials. Numerical experiments verify the main conclusion.

Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

<abstract><p>This paper introduces a weak Galerkin finite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.</p></abstract>

Non-Symmetric Interior Penalty Galerkin Finite Element Method for a Class of Singularly Perturbed Reaction Diffusion Problems with Discontinuous Data

Non-Symmetric Interior Penalty Galerkin Finite Element Method for a Class of Singularly Perturbed Reaction Diffusion Problems with Discontinuous Data

An efficient numerical technique for solving nonlinear singularly perturbed reaction diffusion problem

An efficient numerical technique for solving nonlinear singularly perturbed reaction diffusion problem

Numerical simulation of front dynamics in a nonlinear singularly perturbed reaction–diffusion problem

A new approach to the numerical simulation of the reaction front motion in a nonlinear singularly perturbed partial differential equation of the reaction–diffusion type is proposed. It is shown how the methods of asymptotic analysis allow reducing the statement of the original problem to a problem of a smaller dimension, the numerical solving of which uses methods of diagnostics of the solution’s blow-up. Numerical experiments demonstrate the effectiveness of the proposed approach.

Efficient discretization and preconditioning of the singularly perturbed reaction-diffusion problem

We consider the reaction diffusion problem and present efficient ways to discretize and precondition in the singular perturbed case when the reaction term dominates the equation. Using the concepts of optimal test norm and saddle point reformulation, we provide efficient discretization processes for uniform and non-uniform meshes. We present a preconditioning strategy that works for a large range of the perturbation parameter. Numerical examples to illustrate the efficiency of the method are included for a problem on the unit square.

Supercloseness in a balanced norm of finite element methods on Shishkin and Bakhvalov–Shishkin rectangular meshes for reaction–diffusion problems

We present a convergence analysis for finite element methods of any order, which are applied on Shishkin mesh and Bakhvalov–Shishkin mesh to a singularly perturbed reaction–diffusion problem. A new interpolant is introduced for analysis in the balanced norm. By means of this interpolant and some superconvergence estimations, we prove supercloseness results in the cases of Shishkin mesh and Bakhvalov–Shishkin mesh. Numerical experiments also support these theoretical results.

Optimal fourth-order parameter-uniform convergence of a non-monotone scheme on equidistributed meshes for singularly perturbed reaction–diffusion problems

In this paper, we present an optimal fourth-order parameter-uniform non-monotone scheme on equidistributed meshes for singularly perturbed reaction–diffusion boundary value problems exhibiting boundary layers at both ends of the domain. We discretize the problem using a high-order non-monotone finite difference scheme and prove that the scheme is stable in the maximum norm. The equidistribution of an appropriate monitor function is used to generate the layer-adapted meshes to discretize the problem. The method is proved to be optimal fourth-order uniformly convergent on these equidistributed meshes. Numerical results are presented to validate the theory and to demonstrate the efficiency of the proposed method.

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