Heat transport at the microscale is of vital importace in microtechnology applications. The heat transport equation is different from the traditional heat transport equation since a second order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we develop a hybrid finite element-finite difference (FE-FD) scheme with two levels in time for the three dimensional heat transport equation in a cylindrical thin film with sub-microscale thickness. It is shown that the scheme is unconditionally stable. The scheme is then employed to obtain the temperature rise in a sub-microscale cylindrical gold film. The method can be applied to obtain the temperature rise in any thin films with sub-microscale thickness, where the geometry in the planar direction is arbitrary.

Let $T_{1,n}$ be an $n\times n$ unreduced symmetric tridiagonal matrix with eigenvalues $$\lambda_1<\lambda_2<\cdots<\lambda_n.$$ and $$W_k=\Bigg(\begin{matrix}T_{1,k-1} & 0  \\0&T_{k+1,n}\end{matrix} \Bigg)$$is an $(n-1)\times(n-1)$ submatrix by deleting the $k^{th}$ row and $k^{th}$ column, $k=1,2,\ldots,n$ from $T_n$. Let $$\mu_1\leq\mu_2\leq\cdots\leq\mu_{k-1}$$ be the eigenvalues of $T_{1,k-1}$ and $$\mu_k\leq\mu_{k+1}\leq\cdots\leq\mu_{n-1}$$ be the eigenvalues of $T_{k+1,n}$.

A new inverse eigenvalues problem has put forward as follows: How do we construct an unreduced symmetric tridiagonal matrix $T_{1,n}$, if we only know the spectral data: the eigenvalues of $T_{1,n}$, the eigenvalues of $T_{1,k-1}$ and the eigenvalues of $T_{k+1,n}$?
Namely if we only know the data: $\lambda_1,\lambda_2,\cdots,\lambda_n,\mu_1,\mu_2,\cdots,\mu_{k-1}$ and $\mu_k,\mu_{k+1},\cdots,\mu_{n-1}$ how do we find the matrix $T_{1,n}$? A necessary and sufficient condition and an algorithm of solving such problem, are given in this paper.

In this paper, we investigate the coupling of natural boundary element and finite element methods of exterior initial boundary value problems for hyperbolic equations. The governing equation is first discretized in time, leading to a time-step scheme, where an exterior elliptic problem has to be solved in each time step. Second, a circular artificial boundary $\Gamma_R$ consisting of a circle of radius $R$ is introduced, the original problem in an unbounded domain is transformed into the nonlocal boundary value problem in a bounded subdomain. And the natural integral equation and the Poisson integral formula are obtained in the infinite domain $\Omega_2$ outside circle of radius $R$. The coupled variational formulation is given. Only the function itself, not its normal derivative at artificial boundary $\Gamma_R$, appears in the variational equation, so that the unknown numbers are reduced and the boundary element stiffness matrix has a few different elements. Such a coupled method is superior to the one based on direct boundary element method. This paper discusses finite element discretization for variational problem and its corresponding numerical technique, and the convergence for the numerical solutions. Finally, the numerical example is presented to illustrate feasibility and efficiency of this method.

In this paper, a large class of $n$ dimensional orthogonal and biorthogonal wavelet filters (lowpass and highpass) are presented in explicit expression. We also characterize orthogonal filters with linear phase in this case. Some examples are also given, including nonseparable orthogonal and biorthogonal filters with linear phase.

We present a class of asymptotically optimal successive overrelaxation methods for solving the large sparse system of linear equations. Numerical computations show that these new methods are more efficient and robust than the classical successive overrelaxation method.

  In a composite-step approach, a step $s_k$ is computed as the sum of two components $v_k$ and $h_k$. The normal component $v_k$, which is called the vertical step, aims to improve the linearized feasibility, while the tangential compoent $h_k$, which is also called horizontal step, concentrates on reducing a model of the merit functions. As a filter method, it reduces both the infeasibility and the objective function. This is the same property of these two methods. In this paper, one concerns the composite-step like filter approach. That is, a step is tangential component $h_k$ if the infeasibility is reduced. Or else, $s_k$ is a composite step composed of normal component $v_k$ and tangential component $h_k$.

Based on the work of paper [1], we propose a modified Levenberg-Marquardt algorithm for solving singular system of nonlinear equations $F(x)=0$, where $F(x):R^n\rightarrow R^n$ is continuously differentiable and $F'(x)$ is Lipschitz continuous. The algorithm is equivalent to a trust region algorithm in some sense , and the global convergence result is given. The sequence generated by the algorithm converges to the solution quadratically, if $\|F(x)\|_2$provides a local error bound for the system of nonlinear equations. Numerical results show that the algorithm performs well.

Superconvergence of the mixed finite element methods for 2-d Maxwell equations is studied in this paper. Two order of superconvergent factor can be obtained for the $k$-th Nedelec elements on the rectangular meshes.

A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and space directions. We also obtain a general formula for composition method.

It is well-known that if we have an approximate eigenvalue $\widehat{\lambda}$ of a normal matrix $A$ of order $n$, a good approximation to the corresponding eigenvector $u$ can be computed by one inverse iteration provided the position, say $k_{max}$, of the largest component of $u$ is known. In this paper we give a detailed theoretical analysis to show relations between the eigenvector $u$ and vector $x_k,k=1,\cdots,n$, obtained by simple inverse iteration, i.e., the solution to the system $(A-\widehat{\lambda}I)x=e_k$ with $e_k$ the $k$th column of the identity matrix $I$. We prove that under some weak conditions, the index $k_{max}$ is of some optimal properties related to the smallest residual and smallest approximation error to $u$ in spectral norm and Froenius norm. We also prove that the normalized absolute vector $v=|u|/\|u\|_\infty$ of $u$ can be approximated by the normalized vector of $(\|x_1\|_2,\cdots,\|x_n\|_2)^T$. We also give some upper bounds of $|u(k)|$ for those "optimal" indexes such as Fernando's heuristic for $k_{max}$ without any assumptions. A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of $u$.

In this paper, the solution of back-Euler implicit difference scheme for a semi-linear parabolic equation is proved to converge to the solution of difference scheme for the corresponding semi-linear elliptic equation as $t$ tends to infinity. The long asymptotic behavior of its discrete solution is obtained which is analogous to that of its continuous solution. At last, a few results are also presented for Crank-Nicolson scheme.  

In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in $R^k$ with $k\geq3$.