A singularly perturbed convection–diffusion problem in a half-plane

A singularly perturbed convection–diffusion equation with constant coefficients is considered in a half plane, with Dirichlet boundary conditions. The boundary function has a specified degree of regularity except for a jump discontinuity, or jump discontinuity in a derivative of specified order, at a point. Precise pointwise bounds for the derivatives of the solution are obtained. The bounds show both the strength of the interior layer emanating from the point of discontinuity and the blowup of the derivatives resulting from the discontinuity, and make precise the dependence of the derivatives on the singular perturbation parameter.     

A singularly perturbed reaction–diffusion problem with inhomogeneous environment

Reaction–diffusion problems have been used to describe pattern formation in developmental biology and material science. I study the existence and stability of a singularly perturbed reaction–diffusion problem with inhomogeneous environment in one-dimensional space domain and also high-dimensional space domain.     

On asymptotic expansions in nonlinear,singularly perturbed difference equations ∗

The present paper deals with the problem of constructing and proving asymptotic expansions for nonlinear, singularly perturbed difference equations. New methods for the construction of asymptotic expansions are presented and compared with well-known ones. For the proof of their validity, fundamental principles for the treatment of nonlinear singular perturbation problems are applied, based on the concepts of e-stability, formal asymptotic expansions, matching and asymptotic expansions. The results are derived from a general theory of asymptotic expansions of nonlinear operator equations that has been developed recently by the author.     

A linearised singularly perturbed convection–diffusion problem with an interior layer

A linear time dependent singularly perturbed convection–diffusion problem is examined. The convective coefficient contains an interior layer (with a hyperbolic tangent profile), which in turn induces an interior layer in the solution. A numerical method consisting of a monotone finite difference operator and a piecewise-uniform Shishkin mesh is constructed and analysed. Neglecting logarithmic factors, first order parameter uniform convergence is established.     

A Robin–Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral

In this paper, we consider the Robin–Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral. Using the Faedo–Galerkin method and the linearization method for nonlinear terms, we prove the existence and uniqueness of a weak solution. We also discuss an asymptotic expansion of high order in a small parameter of a weak solution.     

A bilinear approach to the pooling problem†

In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refinery planning models. The main difficulty is that pooling causes an inherent nonlinearity in the otherwise linear models. We shall define the problem by formulating an aggregate mathematical model of a refinery, comment on solution methods for pooling problems that have been presented in the literature, and develop a new method based on convex approximations of the bilinear terms. The method is illustrated on numerical examples     

Qualitative analysis of a singularly perturbed system in ℝ3

We present some qualitative analysis of a singularly perturbed system of ordinary differential equations with two slow variables and one fast variable. The study rests on the method of integral manifolds and its modification in connection with applied problems. The inspection of the system requires studying various types of oscillations. We propose some sufficient conditions for the existence of relaxation oscillations in this system in the case that the slow surface has two folds.     

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

A semilinear reaction–diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter e²{\varepsilon^2} , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult. To address this difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub- and super-solutions. Convergence results are deduced that depend on the relative sizes of e{\varepsilon} and N, where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues in using Newton’s method to compute a discrete solution are discussed.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

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.     

A numerical method for a system of singularly perturbed reaction–diffusion equations

A Dirichlet problem for a system of two coupled singularly perturbed reaction–diffusion ordinary differential equations is examined. A numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed. Numerical results are presented, which illustrate the theoretical results.     

On the convergence of the Dirichlet grid problem with a singularity for a singularly perturbed convection–diffusion equation

The Dirichlet problem for a singulary perturbed convection–diffusion equation in a rectangle when a discontinuity at the flow exit the first derivative of the boundary condition gives rise to an inner layer for the solution. On piecewise-uniform Shishkin grids that condense near regular and characteristic layers, the solution obtained using the classical five-point difference scheme with a directed difference is shown to converge with respect to the small parameter to solve the original problem in the grid norm L _∞ ^h almost with the first order. This theoretical result is confirmed via numerical analysis.     

A second order scheme for a time-dependent,singularly perturbed convection–diffusion equation

We consider a numerical scheme for a one-dimensional, time-dependent, singularly perturbed convection–diffusion problem. The problem is discretized in space by a standard finite element method on a Bakhvalov–Shishkin type mesh. The space error is measured in an L² norm. For the time integration, the implicit midpoint rule is used. The fully discrete scheme is shown to be convergent of order 2 in space and time, uniformly in the singular perturbation parameter.     

A finite element method for two–parameter singularly perturbed problems in 2D

We consider a singularly perturbed elliptic problem with two small parameters posed on the unit square. Based on a decomposition of the solution, we prove uniform convergence of a finite element method in an energy norm. The method uses piecewise bilinear functions on a layer-adapted Shishkin mesh. Numerical results confirm our theoretical analysis. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlinear analytic approach to the splitting theorem of Jacobs–de Leeuw–Glicksberg

In this paper, we study nonlinear analytic methods for linear contractive semigroups in Banach spaces and apply them to the splitting theorem of Jacobs–de Leeuw–Glicksberg. Using these results, we obtain the extension of Lin’s proposition for a group of linear operators to a semigroup.     

A stochastic nonlinear Schrödinger problem in variational formulation

This paper investigates a Schrödinger problem with power-type nonlinearity and Lipschitz-continuous diffusion term on a bounded one-dimensional domain. Using the Galerkin method and a truncation, results from stochastic partial differential equations can be applied and uniform a priori estimates for the approximations are shown. Based on these boundedness results and the structure of the nonlinearity, it follows the unique existence of the variational solution.