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Open Access
January 1, 2010
### Abstract

We consider the class of linear operator equations with operators admitting self-adjoint positive definite and m-accretive splitting (SAS). This splitting leads to an ADI-like iterative method which is equivalent to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of a minimal residual algorithm with Symmetric Gauss-Seidel and polynomial preconditioning is then applied to solve the resulting matrix operator equation. Theoretical analysis shows the convergence of the methods, and upper bounds for the decrease rate of the residual are derived. The convergence of the methods is numerically illustrated with the example of the neutron transport problem in 2-D geometry.

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Open Access
January 1, 2010
### Abstract

Amongst the more exciting phenomena in the field of nonlinear partial differential equations is the Lavrentiev phenomenon which occurs in the calculus of variations. We prove that a conforming finite element method fails if and only if the Lavrentiev phenomenon is present. Consequently, nonstandard finite element methods have to be designed for the detection of the Lavrentiev phenomenon in the computational calculus of variations. We formulate and analyze a general strategy for solving variational problems in the presence of the Lavrentiev phenomenon based on a splitting and penalization strategy. We establish convergence results under mild conditions on the stored energy function. Moreover, we present practical strategies for the solution of the discretized problems and for the choice of the penalty parameter.

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Open Access
January 1, 2010
### Abstract

A second order elliptic problem with highly discontinuous coefficients has been considered. The problem is discretized by two methods: 1) continuous finite element method (FEM) and 2) composite discretization given by a continuous FEM inside the substructures and a discontinuous Galerkin method (DG) across the boundaries of these substructures. The main goal of this paper is to design and analyze parallel algorithms for the resulting discretizations. These algorithms are additive Schwarz methods (ASMs) with special coarse spaces spanned by functions that are almost piecewise constant with respect to the substructures for the first discretization and by piecewise constant functions for the second discretization. It has been established that the condition number of the preconditioned systems does not depend on the jumps of the coefficients across the substructure boundaries and outside of a thin layer along the substructure boundaries. The algorithms are very well suited for parallel computations.

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Open Access
January 1, 2010
### Abstract

The regularizing parameter appearing in some Fredholm integral equations of the second kind is discussed. Theoretical estimates and the results of numerical tests confirming the theoretical expectations are given.

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Open Access
January 1, 2010
### Abstract

In this paper we present a numerical algorithm for solving fuzzy differential equations based on Seikkala’s derivative of a fuzzy process. We discuss in detail a numerical method based on a Runge-Kutta Nystrom method of order three. The algorithm is illustrated by solving some fuzzy differential equations.

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Open Access
January 1, 2010
### Abstract

We propose a novel numerical method for fast and accurate evaluation of the exchange part of the Fock operator in the Hartree-Fock equation which is a (nonlocal) integral operator. Usually, this challenging computational problem is solved by analytical evaluation of two-electron integrals using the “analytically separable” Galerkin basis functions, like Gaussians. Instead, we employ the agglomerated “grey-box” numerical computation of the corresponding six-dimensional integrals in the tensor-structured format which does not require analytical separability of the basis set. The point of our method is a low-rank tensor representation of arising functions and operators on an n×n×n Cartesian grid and the implementation of the corresponding multi-linear algebraic operations in the tensor product format. Linear scaling of the tensor operations, including the 3D convolution product, with respect to the one-dimension grid size n enables computations on huge 3D Cartesian grids thus providing the required high accuracy. The presented algorithm for evaluation of the exchange operator and a recent tensor method for the computation of the Coulomb matrix are the main building blocks in the numerical solution of the Hartree-Fock equation by the tensor-structured methods. These methods provide a new tool for algebraic optimization of the Galerkin basis in the case of large molecules.

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Open Access
January 1, 2010
### Abstract

An implicit flux-corrected transport (FCT) algorithm has been developed for a class of chemotaxis models. The coefficients of the Galerkin finite element discretization has been adjusted in such a way as to guarantee mass conservation and keep the cell density nonnegative. The numerical behaviour of the proposed highresolution scheme is tested on the blow-up problem for a minimal chemotaxis model with singularities. It has also been shown that the results for an Escherichia coli chemotaxis model are in good agreement with the experimental data reported in the literature.