The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in dealing with problems involving anisotropic scatterers. In this paper an adaptive uniaxial PML technique for solving the time harmonic Helmholtz scattering problem is developed. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. The adaptive finite element method based on a posteriori error estimate is proposed to solve the PML equation which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorbing layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method. In particular, it is demonstrated that the PML layer can be chosen as close to one wave-length from the scatterer and still yields good accuracy and efficiency in approximating the far fields.

In this paper, using Lin's integral identity technique, we prove the optimal uniform convergence $O(N_x^{-2}\ln^2N_x+N_y^{-2}\ln^2N_y)$ in the $L^2$-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite 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.

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions. We transform the problem to an initial boundary value problem in dimensionless form. There are two parameters in the coefficients of the resulting linear parabolic partial differential equation. For a range of values of these parameters, the solution of the problem has a boundary or an initial layer. The initial function has a discontinuity in the first-order derivative, which leads to the appearance of an interior layer. We construct analytically the asymptotic solution of the equation in a finite domain. Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in $x$ and at the final time is not affected by the boundary. Also, we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values, while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied. We present numerical computations, which determine experimentally the parameter-uniform rates of convergence. We note that this rate is rather weak, due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.

Pulsatile blood flows in curved atherosclerotic arteries are studied by computer simulations. Computations are carried out with various values of physiological parameters to examine the effects of flow parameters on the disturbed flow patterns downstream of a curved artery with a stenosis at the inner wall. The numerical results indicate a strong dependence of flow pattern on the blood viscosity and inlet flow rate, while the influence of the inlet flow profile to the flow pattern in downstream is negligible.

A system of $m$ (≥ 2) linear convection-diffusion two-point boundary value problems is examined, where the diffusion term in each equation is multiplied by a small parameter $ε$ and the equations are coupled through their convective and reactive terms via matrices $B$ and $A$ respectively. This system is in general singularly perturbed. Unlike the case of a single equation, it does not satisfy a conventional maximum principle. Certain hypotheses are placed on the coupling matrices $B$ and $A$ that ensure existence and uniqueness of a solution to the system and also permit boundary layers in the components of this solution at only one endpoint of the domain; these hypotheses can be regarded as a strong form of diagonal dominance of $B$. This solution is decomposed into a sum of regular and layer components. Bounds are established on these components and their derivatives to show explicitly their dependence on the small parameter $ε$. Finally, numerical methods consisting of upwinding on piecewise-uniform Shishkin meshes are proved to yield numerical solutions that are essentially first-order convergent, uniformly in $ε$, to the true solution in the discrete maximum norm. Numerical results on Shishkin meshes are presented to support these theoretical bounds.

It has been evident that the theory and methods of dynamic derivatives are playing an increasingly important rôle in hybrid modeling and computations. Being constructed on various kinds of hybrid grids, that is, time scales, dynamic derivatives offer superior accuracy and flexibility in approximating mathematically important natural processes with hard-to-predict singularities, such as the epidemic growth with unpredictable jump sizes and option market changes with high uncertainties, as compared with conventional derivatives. In this article, we shall review the novel new concepts, explore delicate relations between the most frequently used second-order dynamic derivatives and conventional derivatives. We shall investigate necessary conditions for guaranteeing the consistency between the two derivatives. We will show that such a consistency may never exist in general. This implies that the dynamic derivatives provide entirely different new tools for sensitive modeling and approximations on hybrid grids. Rigorous error analysis will be given via asymptotic expansions for further modeling and computational applications. Numerical experiments will also be given.

A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on $a$ $priori$ (sequentially) adapted meshes and study its convergence. The scheme on $a$ $priori$ adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which allows us to find $a$ $priori$ a subdomain where the computed solution requires a further improvement. This subdomain is defined by the perturbation parameter $ε$, the step-size of a uniform mesh in $x$, and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations $K$ for improving the solution. To solve the discrete problems aimed at the improvement of the solution, we use uniform meshes on the subdomains. The error of the numerical solution depends weakly on the parameter $ε$. The scheme converges almost $ε$-uniformly, precisely, under the condition $N^{-1}=o\left(ε^{\nu}\right)$, where $N$ denotes the number of nodes in the spatial mesh, and the value $\nu=\nu(K)$ can be chosen arbitrarily small for suitable $K$.

The paper is concerned with strongly nonlinear singularly perturbed boundary value problems in one dimension. The problems are solved numerically by finite-difference schemes on special meshes which are dense in the boundary layers. The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed. For the central scheme, error estimates are derived in a discrete $L^1$ norm. They are of second order and decrease together with the perturbation parameter ε. The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically. Numerical results show ε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.