Most error analyses for numerical integration algorithms specify the space of integrands and then determine the convergence rate for a particular algorithm or the optimal algorithm. This article takes a different perspective of specifying the convergence rate and then finding the largest space of integrands for which the algorithm gives that desired rate. Both worst-case and randomized error analyses are provided.

A three-point vector difference scheme on a infinite interval is considered. Method of reduction of this scheme to a scheme with a finite number of nodes is proposed. Method is based on the extraction of sets of solutions of the difference equation, satisfying the limiting conditions at infinity. The method is applied for numerical solution of an elliptic singularly perturbed problem in a strip. Results of numerical experiments are discussed.

In this paper a suitable Laplacian preconditioner is proposed for the numerical solution of the nonlinear elasto-plastic torsion problem. The aim is to determine the tangential stress in cross-sections under a given torsion, for which the physical model is based on the Saint Venant model of torsion and the single curve hypothesis for the connection of strain and stress. The proposed iterative solution of the arising nonlinear elliptic problem is achieved by combining the advantages of Laplacian preconditioners with the qualitatively favourable aspects of the strong formulation. Error estimate is given for the convergence of the method. Finally, a numerical example is given.

In this paper, we present a regularization method to a degenerate variational inequality of parabolic type arising from American option pricing. Main difficulty in actually analyzing this kind of problem is caused by the presence of a non-smoothing initial value function in the formulation of the problem. We first use a smoothing technique with small parameter $\varepsilon > 0$ to non-smoothing initial value function; and then we derive the error estimates for regularized continuous problem and regularized discrete problem, respectively. Numerical tests are given to confirm our theoretical results.

It is well known that many problems of practical importance in science and engineering have multiple-scale solutions. Moreover, the calculations of numerical methods for these problems is very intensive, even if using some multi-scale procedures. It is therefore important to seek efficient calculation methods. In this paper, superconvergent techniques are used in existing multiscale methods to improve the calculation efficiency. Furthermore, based on comprehensive analysis, the order of the error estimates between the numerical approximation and the exact solution is verified to be improved.

Due to the inherent nature of flip-chip assembly, the solder joints lie beneath the device and therefore are not amenable to visual inspection. Hence, it is important at the design stage to ensure that solder defects such as joint separation or joint shortening do not occur in the assembly. As a first step, the solder joint is modeled using a level-set approach. Unlike conventional front-tracking approaches, the levelset method handles complicated profiles arising from merger/separation of solder joints naturally without user intervention. The model was established to determine the upper and lower limit on optimal solder volume as a function of a specific assembly configuration and is used to avoid such defects.

In this paper singularly perturbed reaction diffusion equations of elliptic and parabolic types have been discussed using wavelet optimized finite difference (WOFD) method based on an interpolating wavelet transform using cubic spline on dyadic points as discussed in [1]. Adaptive feature is performed automatically by thresholding the wavelet coefficients. WOFD [2] works by using adaptive wavelet to generate an irregular grid which is then exploited for the finite difference method. Numerical examples are presented for elliptic and parabolic problems and comparisons have been made using cubic spline and WOFD. The proposed adaptive method is very effective for studying singular perturbation problems in term of adaptive grid generation and CPU time.

We prove an optimal-order error estimate in a weighted energy norm for the Eulerian-Lagrangian localized adjoint method (ELLAM) for unsteady-state advection-diffusion equations with general inflow and outflow boundary conditions. It is well known that these problems admit dynamic fronts with interior and boundary layers. The estimate holds uniformly with respect to the vanishing diffusion coefficient.

The Cauchy problem for the parabolic equation $$\frac{∂u}{∂t} =\frac{∂}{∂x} (k(x,t) \frac{∂u}{∂x}) + f(u,x,,t), x \in R, t > 0,$$ $$u(x,0) = u_0(x), x\in R,$$ is considered. Under conditions $u(x, t) = X(x)T_1(t) + T_2 (t)$, $\frac{∂u}{∂x} \neq 0$, $k(x,t)=k_1(x)k_2(t)$, $f(u,x,t) = f_1(x,t)f_2(u)$, it is shown that the above problem is equivalent to a system of two first-order ordinary differential equations for which exact difference schemes with special Steklov averaging and difference schemes with any order of approximation are constructed on the moving mesh. On the basis of this approach, the exact difference schemes are constructed also for boundary-value problems and multi-dimensional problems. Presented numerical experiments confirm the theoretical results investigated in the paper.

A dual mixed finite element method, for quasi-Newtonian fluid flow obeying to the power law, is constructed and analyzed in [8]. This mixed formulation possesses local (i.e., at element level) conservation properties (conservation of the momentum and the mass) as in the finite volume methods. We propose here an a posteriori error analysis for this mixed formulation.

We consider in this paper the homogeneous 2-D wave equation defined on $\Omega \subset \mathbb{R}^2$. Using the Hilbert Uniqueness Method, one may associate to a suitable fixed subset $\omega \subset \Omega$, the control $v_{\omega}$ of minimal $L^2 (\omega \times (0, T))$-norm which drives to rest the system at a time $T>0$ large enough. We address the question of the optimal position of $\omega$ which minimize the functional $J : \omega \rightarrow ||v_{\omega}||_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^1(\Omega)$, we express the shape derivative of $J$ as a curvilinear integral on $∂\omega \times (0,T)$ independently of any adjoint solution. This expression leads to a descent direction and permits to define a gradient algorithm efficiently initialized by the topological derivative associated to $J$. The numerical approximation of the problem is discussed and numerical experiments are presented in the framework of the level set approach. We also investigate the well-posedness of the problem by considering its relaxation.