A Finite Element Method for Singularly Perturbed Reaction—diffusion Problems

A finite element method is proposed for the sing ularly perturbed reaction-diffusion problem.An optimal error bound is derived,independent of the perturbation parameter.     

The hp finite element method for singularly perturbed problems in nonsmooth domains

We consider the numerical approximation of singularly perturbed elliptic boundary value problems over nonsmooth domains. We use a decomposition of the solution that contains a smooth part, a corner layer part and a boundary layer part. Explicit guidelines for choosing mesh‐degree combinations are given that yield finite element spaces with robust approximation properties. In particular, we construct an hp finite element space that approximates all components uniformly, at a near exponential rate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 63–89, 1999     

Uniform Convergence Analysis for Singularly Perturbed Elliptic Problems with Parabolic Layers

In this paper,using Lin's integral identity technique,we prove the optimal uniform convergence θ(N_x~(-2)In~2N_x N_y~(-2)In~2N_y) in the L~2-norm for singularly per- turbed problems with parabolic layers.The error estimate is achieved by bilinear fi- nite elements on a Shishkin type mesh.Here N_x and N_y are the number of elements in the x- and y-directions,respectively.Numerical results are provided supporting our theoretical analysis.     

流线扩散有限元方法在分层网格上的收敛性分析

本文在分层网格上分析了采用线性元的流线扩散有限元方法求解一维对流扩散型奇异摄动问题的一致收敛性.在ε≤N~(-1)的前提下,可以证明在SD范数下的一致误差估计为O(N~(-1)(log 1/ε)~2)在数值算例部分对理论结果进行了验证.     

奇异摄动问题在Bakhvalov-Shishkin 网格上的流线扩散有限元逼近

本文采用线性插值的流线扩散有限元在Bakhvalov-Shishkin网格上求解一维对流扩散型的奇异摄动问题. 在ε ≤ N^-1的前提下,可以得到,关于扰动参数ε 是一致收敛的. 在离散的SD范数下,其u-u_I的误差阶提高到N^-2,u-u_h的误差阶达到N^-2（lnN）^0.5. 最后,通过数值算例,验证了理论分析.     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

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

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.     

Parameter uniform approximations for time‐dependent reaction‐diffusion problems

Using discrete Green's functions techniques, we present a classification of fitted mesh methods for time‐dependent reaction diffusion problems, based on the analyses of Linß (Linß, Numer Algor 40 (2005), 23–32) for the analogous steady‐state problem and of Kopteva (Kopteva, Computing 66 (2001), 179–197) for time‐dependent convection‐diffusion problems. As examples of how to apply the analysis, we derive error estimates for the fitted meshes of Shishkin and Bakhvalov, and provide supporting numerical results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

修正特征线有限元法的局部逐点误差估计

本文考虑抛物型对流占优的扩散问题的集中质量特征有限元方法，给出了与小粘性参数无关的局部逐点误差估计，即此格式在边界层外关于小粘性一致收敛．     

Nonstationary three-dimensional contrasting structures

Three-dimensional contrasting structures (CS) occurring in nonlinear diffusion problems with generation are considered assuming that the generation coefficient depends on the concentration. Conditions under which a CS occupying a nonconvex domain in the three-dimensional space disintegrates into several isolated parts in the course of evolution are formulated. This property distinguishes three-dimensional CSs from the two-dimensional ones; the surface of the latter does not change its connectivity until the structure completely disappears.     

Generating Layer-Adapted Meshes Using Mesh Partial Differential Equations

We present a new algorithm for generating layer-adapted meshes for the finite element solution of singularly perturbed problems based on mesh partial differential equations (MPDEs). The ultimate goal is to design meshes that are similar to the well-known Bakhvalov meshes, but can be used in more general settings: specifically two-dimensional problems for which the optimal mesh is not tensor-product in nature. Our focus is on the efficient implementation of these algorithms, and numerical verification of their properties in a variety of settings. The MPDE is a nonlinear problem, and the efficiency with which it can be solved depends adversely on the magnitude of the perturbation parameter and the number of mesh intervals. We resolve this by proposing a scheme based on $h$-refinement. We present fully working FEniCS codes [Alnaes et al., Arch. Numer. Softw., 3 (100) (2015)] that implement these methods, facilitating their extension to other problems and settings.     

Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables

In an unbounded (with respect to x and t) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter ε² multiplying the higher order derivative is considered. The parameter ε takes arbitrary values in the half-open interval (0, 1]. To solve this problem, difference schemes on grids with an infinite number of nodes (formal difference schemes) are constructed that converge ε-uniformly in the entire unbounded domain. To construct these schemes, the classical grid approximations of the problem on the grids that are refined in the boundary layer are used. Schemes on grids with a finite number of nodes (constructive difference schemes) are also constructed for the problem under examination. These schemes converge for fixed values of ε in the prescribed bounded subdomains that can expand as the number of grid points increases. As ε → 0, the accuracy of the solution provided by such schemes generally deteriorates and the size of the subdomains decreases. Using the condensing grid method, constructive difference schemes that converge ε-uniformly are constructed. In these schemes, the approximation accuracy and the size of the prescribed subdomains (where the schemes are convergent) are independent of ε and the subdomains may expand as the number of nodes in the underlying grids increases.     

ON THE hp FINITE ELEMENT METHOD FOR THE ONE DIMENSIONAL SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS

In this work, a singularly perturbed two-point boundary value problem of convection-diffusion type is considered. An hp version finite element method on a strongly graded piecewise uniform mesh of Shishkin type is used to solve the model problem. With the analytic assumption of the input data, it is shown that the method converges exponentially and the convergence is uniformly valid with respect to the singular perturbation parameter.     

Singularly perturbed two-dimensional parabolic problem in the case of intersecting roots of the reduced equation

The singularly perturbed parabolic equation ?u _t + ε²Δu ? f(u, x, ε) = 0, x ∈ D ? ?², t > 0 with Robin conditions on the boundary of D is considered. The asymptotic stability as t → ∞ and the global domain of attraction are analyzed for the stationary solution whose limit as ε → 0 is a nonsmooth solution to the reduced equation f(u, x, 0) = 0 that consists of two intersecting roots of this equation.