An unstructured mesh finite volume discretisation method for simulating diffusion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume finite-element method and it retains the local continuity of the flux at the control volume faces. A least squares function reconstruction technique together with a new flux decomposition strategy is used to obtain an accurate flux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it significantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.  

By making use of the quotient singular value decomposition (QSVD) of a matrix pair, this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the general solutions of the linear matrix equation $AXA^T+BYB^T=C$ with the unknown $X$ and $Y$, which may be both symmetric, skew-symmetric, nonnegative definite , positive definite or some cross combinations respectively. Also, the solutions of some optimal problems are derived.

For a model elliptic boundary value problem we will prove that on strongly regular families of uniform tetrahedral partitions of a pohyhedral domain, the gradient of the quadratic finite element approximation is superclose to the gradient of the quadratic Lagrange interpolant of the exact solution. This supercloseness will be used to construct a post-processing that increases the order of approximation to the gradient in the global $L^2$-norm.

This paper is concerned with the numerical solution of functional-differential and functional equations which include functional-differential equations of neutral type as special cases. The adaptation of general linear methods is considered. It is proved that A-stable general linear methods can inherit the asymptotic stability of underlying linear systems. Some general results of numerical stability are also given.  

In this paper we consider continuous-time and discrete-time waveform relaxation methods for general nonlinear integral-differential-algebraic equations. For the continuous-time case we derive the convergence condition of the iterative methods by invoking the spectral theory on the resulting iterative operators. By using the implicit difference forms, namely the backward-differentiation formulae, we also yield the convergence condition of the discrete waveforms. Numerical experiments are provided to illustrate the theoretical work reported here.

In order to construct closed surfaces with continuous unit normal, we introduce a new spline space on an arbitrary closed mesh of three-sided faces. Our approach generalizes an idea of Goodman and is based on the concept of 'Geometric continuity' for piecewise polynomial parametrizations. The functions in the spline space restricted to the faces are cubic triangular polynomials. A basis of the spline space is constructed of positive functions which sum to 1. It is also shown that the space is suitable for interpolating data at the midpoints of the faces.

In this paper,we will prove the derivative of tetrahedral quadratic finite element approximation is superapproximate to the derivative of the quadratic Lagrange interpolant of the exact solution in the $L^{\infty}$-norm, which can be used to enhance the accuracy of the derivative of tetrahedral quadratic finite element approximation to the derivative of the exact solution.  

It has been observed that the residual polynomials resulted from successive restarting cycles of GMRES($m$) may differ from one another meaningfully. In this paper, it is further shown that the polynomials can complement one another harmoniously in reducing the iterative residual. This characterization of GMRES($m$) is exploited to formulate an efficient hybrid iterative scheme, which can be widely applied to existing hybrid algorithms for solving large nonsymmetric systems of linear equations. In particular, a variant of the hybrid GMRES algorithm of Nachtigal, Reichel and Trefethen (1992) is presented. It is described how the new algorithm may offer significant performance improvements over the original one.

We study the behavior of some polynomial interior-point algorithms for solving random linear programming (LP) problems. We show that the expected and anticipated number of iterations of these algorithms is bounded above by $O(n^{1.5})$. The random LP problem is Todd's probabilistic model with the Cauchy distribution.

In the present paper, the authors discuss the locking phenomenon of the finite element method for three-dimensional elasticity as the Lamé constant $\lambda \rightarrow\infty$. Three kinds of finite elements are proposed and analyzed to approximate the three-dimensional elasticity with pure displacement boundary condition. Optimal order error estimates which are uniform with respect to $\lambda\in (0,+\infty)$ are obtained for three schemes. Furthermore, numerical results are presented to show that, our schemes are locking-free and and the trilinear conforming finite element scheme is locking.