Perturbed Reaction-diffusion Equations Research Articles

Superconvergence analysis of the conforming discontinuous Galerkin method on a Bakhvalov-type mesh for singularly perturbed reaction–diffusion equation

The conforming discontinuous Galerkin (CDG) method maximizes the utilization of all degrees of freedom of the discontinuous Pk polynomial to achieve a convergence rate two orders higher than its counterpart conforming finite element method employing continuous Pk element. Despite this superiority, there is little theory of the CDG methods for singular perturbation problems. In this paper, superconvergence of the CDG method is studied on a Bakhvalov-type mesh for a singularly perturbed reaction–diffusion problem. For this goal, a pre-existing least squares method has been utilized to ensure better approximation properties of the projection. On the basis of that, we derive superconvergence results for the CDG finite element solution in the energy norm and L2-norm and obtain uniform convergence of the CDG method for the first time.

Supercloseness and post-processing of finite element method in a balanced norm for singularly perturbed reaction-diffusion equation

Supercloseness and post-processing of finite element method in a balanced norm for singularly perturbed reaction-diffusion equation

Robust Estimates in Balanced Norms for Singularly Perturbed Reaction Diffusion Equations Using Graded Meshes

Robust Estimates in Balanced Norms for Singularly Perturbed Reaction Diffusion Equations Using Graded Meshes

Fitted finite difference technique for singularly perturbed reaction diffusion equations of third order with integral boundary condition

AbstractWe have considered a third order singularly perturbed reaction diffusion equations with integral boundary condition. On a Shishkin mesh, a numerical technique based on a finite difference scheme is provided. The proposed approach is almost first order convergent. In the discrete norm, an error estimate is obtained. Numerical examples are provided to validate the theoretical estimates.

A robust numerical technique for weakly coupled system of parabolic singularly perturbed reaction–diffusion equations

A robust numerical technique for weakly coupled system of parabolic singularly perturbed reaction–diffusion equations

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>

AN EFFICIENT PARAMETER UNIFORM SPLINE-BASED TECHNIQUE FOR SINGULARLY PERTURBED WEAKLY COUPLED REACTION-DIFFUSION SYSTEMS

A parameter-uniform numerical scheme for a system of weakly coupled singularly perturbed reaction-diffusion equations of arbitrary size with appropriate boundary conditions is investigated. More precisely, quadratic $B$-spline basis functions with an exponentially graded mesh are used to solve a $\ell\times\ell$ system whose solution exhibits parabolic (or exponential) boundary layers at both endpoints of the domain. A suitable mesh-generating function is used to generate the exponentially graded mesh. The decomposition of the solution into regular and singular components is obtained to provide error estimates. A convergence analysis is addressed, which shows a uniform convergence of the second order. To validate the theoretical findings, two test problems are solved numerically.

A parameter-uniform weak Galerkin finite element method for a coupled system of singularly perturbed reaction-diffusion equations

The aim of this paper to investigate a weak Galerkin finite element method (WG-FEM) for solving a system of coupled singularly perturbed reaction-diffusion equations. Each equation in the system has perturbation parameter of different magnitude and thus, the solutions will exhibit two distinct but overlapping boundary layers near each boundary of the domain. The proposed method is applied to the coupled system on Shishkin mesh to solve the problem theoretically and numerically. Elimination of the interior unknowns efficiently from the discrete solution system reduces the degrees of freedom and, thus the number of unknown in the discrete solution is comparable with the standard finite element scheme. The stability and error analysis of the proposed method on the Shishkin mesh are presented. We show that the method convergences of order O(N?k lnk N) in the energy norm, uniformly with respect to the perturbation parameter. Moreover, the optimal convergence rate of O(N?(k+1)) in the L2-norm and the superconvergence rate of O((N?2k ln2k N) in the discrete L?-norm is observed numerically. Finally, some numerical experiments are carried out to verify numerically theory.

Hp-FEM for reaction–diffusion equations. II: robust exponential convergence for multiple length scales in corner domains

Abstract In bounded, polygonal domains $\varOmega \subset {{\mathbb {R}}}^2$ with Lipschitz boundary $\partial \varOmega $ consisting of a finite number of Jordan curves admitting analytic parametrizations, we analyze $hp$-FEM discretizations of linear, second-order, singularly perturbed reaction–diffusion equations on so-called geometric boundary layer meshes. We prove, under suitable analyticity assumptions on the data, that these $hp$-FEM afford exponential convergence in the natural ‘energy’ norm of the problem, as long as the geometric boundary layer mesh can resolve the smallest length scale present in the problem. Numerical experiments confirm the robust exponential convergence of the proposed $hp$-FEM.

Exponential convergence of hp FEM for spectral fractional diffusion in polygons

For the spectral fractional diffusion operator of order 2s, s in (0,1), in bounded, curvilinear polygonal domains varOmega subset {{mathbb {R}}}^2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm {mathbb {H}}^s(varOmega ). The first hp discretization is based on writing the solution as a co-normal derivative of a 2+1-dimensional local, linear elliptic boundary value problem, to which an hp-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations invarOmega . Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in varOmega , exponential convergence rates for solutions uin {mathbb {H}}^s(varOmega ) of mathcal {L}^s u = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towardspartial varOmega . The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of mathcal {L}^{-s} combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations invarOmega . The present analysis for either approach extends to (polygonal subsets {widetilde{mathcal {M}}} of) analytic, compact 2-manifolds {mathcal {M}}, parametrized by a global, analytic chart chi with polygonal Euclidean parameter domain varOmega subset {{mathbb {R}}}^2. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced.

Analysis of a Fitted Finite Difference Scheme for a Semilinear System of Singularly Perturbed Reaction-Diffusion Equations Having Discontinuous Source Term

In this paper, a system of an arbitrary number of singularly perturbed semilinear differential equations of reaction-diffusion type having discontinuous source term is examined, for the case in which the diffusion parameters associated with each equation of the considered system are assumed to be different in magnitude. In addition to the occurrence of the boundary layers, the solution exhibits the overlapping and interacting internal layers due to the existence of discontinuity in the data. A central finite difference scheme is used in conjunction with a suitable piecewise uniform Shishkin mesh and a Bakhvalov mesh, to construct the numerical method. Using discrete Green’s function technique, the proposed numerical scheme has been proved to be an almost second-order of uniform convergent for the Shishkin mesh and second-order of uniform convergent for the Bakhvalov mesh, independent of the perturbation parameters. Numerical test examples are presented to demonstrate the performance of the numerical scheme.

Slow localized patterns in singularly perturbed two-component reaction–diffusion equations

Localized patterns in singularly perturbed reaction–diffusion equations typically consist of slow parts, in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system, alternated by fast excursions, in which the solution jumps from one slow manifold to another, or back to the original slow manifold. In this paper we consider the existence and stability of stationary and travelling localized patterns that do not exhibit such jumps, i.e. that are completely embedded in a slow manifold of the singularly perturbed spatial dynamical system. These ‘slow patterns’ have rarely been considered in the literature, for two reasons: (i) in the classical Gray–Scott/Gierer–Meinhardt type models that dominate the literature, the flow on the slow manifold is typically linear and thus cannot exhibit homoclinic pulse or heteroclinic front solutions; (ii) the slow manifolds occurring in the literature are typically ‘vertical’, i.e. given by u ≡ u 0, where u is the fast variable, so that the stability problem is determined by a simple (decoupled) scalar equation. The present research concerns a general system of singularly perturbed reaction–diffusion equations and is motivated by several explicit ecosystem models that do give rise to non-vertical normally hyperbolic slow manifolds on which the flow may exhibit both homoclinic and heteroclinic orbits that correspond to either stationary or travelling localized slow patterns. The associated spectral stability problems are at leading order given by a nonlinear, but scalar, eigenvalue problem with Sturm–Liouville characteristics and we establish that homoclinic pulse patterns are typically unstable, while heteroclinic fronts can either be stable or unstable. However, we also show that homoclinic pulse patterns that are asymptotically close to a heteroclinic cycle may be stable. This result is obtained by explicitly determining the leading order approximations of four critical asymptotically small eigenvalues. By this analysis, that involves several orders of magnitude in the small parameter, we also obtain full control over the nature of the bifurcations—saddle-node, Hopf, global, etc—that determine the existence and stability of the (stationary and/or travelling) heteroclinic fronts and/or homoclinic pulses. Finally, we show that heteroclinic orbits may correspond to stable (slow) interfaces in two-dimensional space, while the homoclinic pulses must be unstable as localized stripes, even when they are stable in one space dimension.

A higher-order finite difference method for two-dimensional singularly perturbed reaction-diffusion with source-term-discontinuous problem

This paper considers a two-dimensional singularly perturbed reaction-diffusion equation with a discontinuous source term. Due to this discontinuity, interior, corner, and boundary layers appear in the solution for adequately small values of the perturbation parameter ϵ. To achieve a decent estimate of the solution, we construct a numerical approach adopting an efficient hybrid finite difference method that includes a proper layer adapted piece-wise uniform Shishkin mesh. Further, we prove that the hybrid finite difference method is almost second-order uniformly convergent with respect to the perturbation parameter. We have implemented our method to test examples. Numerical results are verifying the theoretical results.

Rate of convergence of global attractors for some perturbed reaction-diffusion equations under smooth perturbations of the domain

Rate of convergence of global attractors for some perturbed reaction-diffusion equations under smooth perturbations of the domain

Singularly perturbed reaction–diffusion problems on a k‐star graph

Singularly perturbed reaction–diffusion equations on a star graph (having k + 1 nodes and k edges) resulting in a system with k individual partial differential equations along the edges with coupling conditions at the common junction are presented. In the singular limit, as the diffusion parameter tends to zero, possibly individually along each edge, boundary layers may occur at the multiple nodes as well as at the simple nodes. Numerically, the proposed equations are solved using central finite difference schemes on properly extended Shishkin meshes. Error estimates are discussed and validated by solving a test problem on a graph with three edges (tripod). A more general graph problem with eight edges and three connecting nodes has also been solved numerically.

A Cartesian grid based tailored finite point method for reaction-diffusion equation on complex domains

This paper presents a Cartesian grid based tailored finite point method (TFPM) for singularly perturbed reaction-diffusion equation on complex domains. The method is incorporated with the kernel-free boundary integral algorithm, where the semi-discrete boundary value problems after time integration are reformulated into corresponding Fredholm boundary integral equations (BIEs) of the second kind, however with no algorithmic dependence on the exact analytical expression for the kernels of integrals. The BIEs are iteratively solved by the GMRES method while integral evaluation during each iteration resorts to solving an equivalent interface problem, which in practice is achieved by a series of manipulations in the framework of TFPM including discretization, correction, solution, and interpolation. The proposed method has second-order accuracy for the reaction-diffusion equation as demonstrated by the numerical examples.

Multi-spike Patterns in the Gierer–Meinhardt System with a Nonzero Activator Boundary Flux

The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction-diffusion equations has been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems we consider the effect of inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer-Meinhardt reaction-diffusion system. We show that these boundary conditions necessitate the formation of spikes that concentrate in a boundary layer near the domain boundaries. Using the method of matched asymptotic expansions we construct boundary layer spikes and derive a new class of shifted Nonlocal Eigenvalue Problems for which we rigorously prove partial stability results. Moreover by using a combination of asymptotic, rigorous, and numerical methods we investigate the structure and linear stability of selected one- and two-spike patterns. In particular we find that inhomogeneous Neumann boundary conditions increase both the range of parameter values over which asymmetric two-spike patterns exist and are stable.

The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation

We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i)~the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii)~the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.

Difference Scheme on a Non-Uniform Mesh for Singularly Perturbed Reaction Diffusion Equations with Integral Boundary Condition

In this paper we consider singularly perturbed reaction diffusion equations with integral boundary condition. A numerical method based on finite difference scheme on Shishkin mesh is presented. This method is proved to be almost second order convergent. An error estimate is derived in the discrete norm. Numerical results are presented, which validate the theoretical results.

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.