Abstract:
Second order elliptic equation is a class of mathematical model for scientific computing, such as convex-diffusion, oil-reservoir simulation, etc. Based on intrinsic symmetrizable property, a new concept on positively symmetrizable matrix is proposed in this paper. We point that for such kind of equation systems, it is possible to adopt special preconditioning CG algorithm, e.g. 1]-3], instead of the usual iteration procedure for general non-symmetry systems, such as GMRES 3]-4] ) BiCGSTAB 5]. Numerical tests show the new algorithm is effective for solving this kind of second order elliptic discrete systems.

Abstract:
The discrete Fourier analysis on the 30°-60°-90° triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G2, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of m-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.

Abstract:
Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.

Abstract:
Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on $[-1,1]^2$, as well as new results on $[-1,1]^3$. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on $n^3/4 +\CO(n^2)$ nodes of a cubature formula on $[-1,1]^3$.

Abstract:
A discrete Fourier analysis associated with translation lattices is developed recently by the authors. It permits two lattices, one determining the integral domain and the other determining the family of exponential functions. Possible choices of lattices are discussed in the case of lattices that tile $\RR^2$ and several new results on cubature and interpolation by trigonometric, as well as algebraic, polynomials are obtained.

Abstract:
The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.

Abstract:
Some testing results on DAWNING-1000, Paragon and workstation cluster are described in this paper. On the home-made parallel system DAWNING-1000 with 32 computational processors, the practical performance of 1.117 Gflops and 1.58 Gflops has been measured in solving a dense linear system and doing matrix multiplication, respectively. The scalability is also investigated. The importance of designing efficient parallel algorithms for evaluating parallel systems is emphasized.

Abstract:
By using the theroy of extremal signature proposed by Rivlin and Shapiro,we prove that some of the Chebyshev polynomials in two variables of the first kind presented in are exactly the polynomials of least deviation from zero on the so-called Steiner's domain. Based on the sets of critial points of these Chebysev polynomials,we present several cubature formulas with certain degree.

Abstract:
In this paper, we propose a new set of orthogonal basis functions in the arbitrary triangular domain. At first, we generalize the 1-D Sturm-Liouville equation to the arbitrary triangular domain on a barycentric coordinate, and derive a set of complete orthogonal basis functions on this domain. Secondly, we analyze the symmetry and periodicity property of these functions and classify them into four classes. At last, we show some of the visualization results of these basis functions.

Abstract:
In this paper we propose so-called coupled 4-point difference schemes for the Laplacian operator over a class of parallel hexagon partitions of the plane. The scheme exhibits global second order accuracy, despite its first order local truncation error. Based on the fact the parallel hexagonal grid can be decoupled into two sets of 3-direction triangular grids, a detailed proof for the second order behavior is given and the corresponding fast solver for Helmholtz equation is also studied. Numerical examples are provided to varify our analysis.