This paper proposes a robust finite element method for a three-dimensional fourth-order elliptic singular perturbation problem. The method uses the three-dimensional Morley element and replaces the finite element functions in the part of bilinear form corresponding to the second-order differential operator by a suitable approximation. To give such an approximation, a convergent nonconforming element for the second-order problem is constructed. It is shown that the method converges uniformly in the perturbation parameter.

We study galvanic currents on a heterogeneous surface. In electrochemistry, the oxidation-reduction reaction producing the current is commonly modeled by a nonlinear elliptic boundary value problem. The boundary condition is of exponential type with periodically varying parameters. We construct an approximation by first homogenizing the problem, and then linearizing about the homogenized solution. This approximation is far more accurate than both previous approximations or direct linearization. We establish convergence estimates for both the two- and three-dimensional case and provide two-dimensional numerical experiments.

We consider the unbalanced Procrustes problem with an orthonormal constraint on solutions: given matrices $A\in \mathcal{R}^{n\times n}$ and $B\in \mathcal{R}^{n\times k}$, $n>k$, minimize the residual $\|AQ-B\|_F$ over the Stiefel manifold of orthonormal matrices. Theoretical analysis on necessary conditions and sufficient conditions for optimal solutions of the unbalanced Procrustes problem is given.

In this paper, we consider the static output feedback (SOF) ${\cal H}_\infty$-synthesis problem posed as a nonlinear semi-definite programming (NSDP) problem. Two numerical algorithms are developed to tackle the NSDP problem by solving the corresponding Karush-Kuhn-Tucker first-order necessary optimality conditions iteratively. Numerical results for various benchmark problems illustrating the performance of the proposed methods are given.

We prove that any linear multi-step method $G_1^{\tau}$ of the form $$\sum^{m}_{k=0}\alpha _k Z_{k}=\tau \sum^{m}_{k=0}\beta _k J^{-1}\nabla H(Z_k)$$ with odd order $u$ ($u\ge 3$) cannot be conjugate to a symplectic method $G_2^{\tau}$ of order $w$ ($w\ge u$) via any generalized linear multi-step method $G_3^{\tau}$ of the form $$\sum^m_{k=0} \alpha_k Z_k = \tau\sum^m_{k=0} \beta_k J^{-1}\nabla H(\sum^m_{l=0}\gamma_{kl}Z_l).$$ We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when $G_3^{\tau}$ is a more general operator.

In this paper, we consider the higher divided difference of a composite function $f(g(t))$ in which $g(t)$ is an $s$-dimensional vector. By exploiting some properties from mixed partial divided differences and multivariate Newton interpolation, we generalize the divided difference form of Faà di Bruno's formula with a scalar argument. Moreover, a generalized Faà di Bruno's formula with a vector argument is derived.

The convergence properties of Newton's method for systems of equations with constant rank derivatives are studied under the hypothesis that the derivatives satisfy some weak Lipschitz conditions. The unified convergence results, which include Kantorovich type theorems and Smale's point estimate theorems as special cases, are obtained.

A fourth-order operator marching method for the Helmholtz equation in a waveguide is developed in this paper. It is derived from a  new fourth-order exponential integrator for linear evolution equations.  The method improves the second-order accuracy associated with the widely used step-wise coupled mode method where the waveguide is  approximated by segments that are uniform in the propagation direction. The Helmholtz equation is solved using a one-way reformulation based on the  Dirichlet-to-Neumann map. An alternative version closely related to  the coupled mode method is also given. Numerical results clearly indicate that the method is more accurate than the coupled mode method while the required computing effort is nearly the same.

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.