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January 1, 2012
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In this work we consider preconditioned iterative solution methods for numerical simulations of multiphase flow problems, modelled by the Cahn-Hilliard equation. We focus on diphasic flows and the construction and efficiency of a preconditioner for the algebraic systems arising from finite element discretizations in space and the θ-method in time. The preconditioner utilises to a full extent the algebraic structure of the underlying matrices and exhibits optimal convergence and computational complexity properties. Various numerical experiments, including large scale examples, are presented as well as performance comparisons with other solution methods.

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January 1, 2012
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We consider the weak analogues of certain strong stochastic numerical schemes, namely an Adams-Bashforth scheme and a semi-implicit leapfrog scheme. We show that the weak version of the Adams-Bashforth scheme converges weakly with order 2, and the weak version of the semi-implicit leapfrog scheme converges weakly with order 1. We also note that the weak schemes are computationally simpler and easier to implement than the corresponding strong schemes, resulting in savings in both programming and computational effort.

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January 1, 2012
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In this paper, we consider an iterative method for the approximate solution of the nonlinear ill-posed operator equation Tx = y. The iteration procedure converges quadratically to the unique solution of the equation for the regularized approximation. It is known that (Tautanhahn (2002)) this solution converges to the solution of the given ill-posed operator equation. The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate.

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January 1, 2012
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Based on the functional-discrete technique (FD-method), an algorithm for solving eigenvalue transmission problems with a discontinuous flux and the integrable potential is developed. The case of the potential as a function belonging to the functional space L_1 is studied for both linear and nonlinear eigenvalue problems. The sufficient conditions providing superexponential convergence rate of the method were obtained. Numerical examples are presented to support the theory. Based on the numerical examples and the convergence results, conclusion about analytical properties of eigensolutions for nonself-adjoint differential operators is made.

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January 1, 2012
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This paper is concerned with the construction and analysis of a parallel preconditioner for a FETI-DP system of equations arising from the nonconforming Crouzeix-Raviart finite element discretization of a model elliptic problem of second order with discontinuous coefficients. We show that the condition number of the preconditioned problem is independent of the coefficient jumps, and the preconditioner is quasi optimal.

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January 1, 2012
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In this paper, we utilize a matrix formulation for the heat equation with nonlocal boundary conditions. We give details of the method and verify the existence of a solution for the given equation under a new suitable condition. A simple and efficient algorithm is presented for the numerical results illustrating the accuracy of the method.

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January 1, 2012
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Finite difference approximations to multi-asset American put option price are considered. The assets are modelled as a multi-dimensional diffusion process with variable drift and volatility. Approximation error of order one quarter with respect to the time discretisation parameter and one half with respect to the space discretisation parameter is proved by reformulating the corresponding optimal stopping problem as a solution of a degenerate Hamilton-Jacobi-Bellman equation. Furthermore, the error arising from restricting the discrete problem to a finite grid by reducing the original problem to a bounded domain is estimated.