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We consider solution techniques for the coupling of Darcy and Stokes flow problems. The study was motivated by the simulation of the interaction between channel flow and subsurface water flow for realistic data and arbitrary interfaces between the two different flow regimes. Here, the emphasis is on the efficient iterative solution of the coupled problem based on efficient solvers for the discrete...
We study the numerical approximation of solutions for parabolic integro-differential equations (PIDE). Similar models arise in option pricing, to generalize the Black–Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Due to the non-local nature of the integral term, unconditionally stable implicit difference schemes are...
In this paper, we use the group inverse to characterize the quotient convergence of an iterative method for solving consistent singular linear systems, when the matrix index equals one. Next, we show that for stationary splitting iterative methods, the convergence and the quotient convergence are equivalent, which was first proved in [7]. Lastly, we propose a (multi-)splitting iterative method A...
In this article the Marchenko integral equations leading to the solution of the inverse scattering problem for the 1-D Schrödinger equation on the line are solved numerically. The linear system obtained by discretization has a structured matrix which allows one to apply FFT based techniques to solve the inverse scattering problem with minimal computational complexity. The numerical results agree...
We provide explicit expressions for both mixed and componentwise structured condition numbers for several classes of structured rectangular matrices: upper triangular, Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Such expressions for many other classes of matrices can be similarly derived. Mathematics Subject Classification (2000): 15A12, 65F35
In the present note we prove convergence results for over-iterates of certain (generalized) Bernstein-Stancu operators. Similar assertions were obtained in [11]. However, our approach is different in the sense that it uses the spectrum of the operators involved. It is therefore possible to make global statements on [0, 1]. Keywords Bernstein-Stancu operators, eigenvalues, eigenfunctions, iterates...
For single splittings of Hermitian positive definite matrices, there are well-known convergence and comparison theorems. This paper gives new convergence and comparison results for double splittings of Hermitian positive definite matrices. Keywords: Hermitian positive definite matrix; convergence theorem; comparison theorem; double splitting Mathematics Subject Classification (2000): 65F10
The two-dimensional Navier–Stokes equations, when subject to non-standard boundary conditions which involve the normal component of the velocity and the vorticity, admit a variational formulation with three independent unknowns, the vorticity, velocity and pressure. We propose a discretization of this problem by spectral element methods. A detailed numerical analysis leads to optimal error estimates...
An optimal control problem for a two-dimensional elliptic equation with pointwise control constraints is investigated. The domain is assumed to be polygonal but non-convex. The corner singularities are treated by a priori mesh grading. Approximations of the optimal solution of the continuous optimal control problem are constructed by a projection of the discrete adjoint state. It is proved that these...
This paper investigates the effects of matrix summability methods on the A-statistical approximation of sequences of positive linear operators defined on the space of all 2π-periodic and continuous functions on the whole real axis. The two main tools used in this paper are A-statistical convergence and the modulus of continuity. Keywords: Regular infinite matrices, A-statistical convergence, rates...
We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in $C^{2p-2}\left( \mathbb{R}^{n+1}\right) $ , polyharmonic of order p on each strip $\left( j,j+1\right) \times\mathbb{R}^{n}$ , , and periodic in its last n variables, whose restriction to the parallel hyperplanes...
In this paper, we give a new (and simpler) stability proof for a cell-centered colocated finite volume scheme for the 2D Stokes problem, which may be seen as a particular case of a wider class of methods analyzed in [10]. The definition of this scheme involves two grids. The coarsest is a triangulation of the computational domain by acute-angled simplices, called clusters. The control volumes grid...
The method of Shepard is an efficient method for interpolation of very large scattered data sets; unfortunately, it has poor reproduction qualities and high computational cost. In this paper we introduce a new operator which diminishes these drawbacks. This operator results from the combination of the Shepard operator with a new interpolation operator, recently proposed by Costabile and Dell’Accio,...
In this paper, we study the approximation of matrix-exponential distributions by Coxian distributions. Based on the spectral polynomial algorithm, we develop an algorithm for computing Coxian representations of Coxian distributions that are approximations of matrix-exponential distributions. As a specialization, we show that phase-type (PH) distributions can be approximated by Coxian distributions...
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