Summary In this paper, we consider efficient algorithms for solving the algebraic equation Aαboldu=boldf, 0<α<1, where scriptA is a properly scaled symmetric and positive definite matrix obtained from finite difference or finite element approximations of second‐order elliptic problems in Rd, d=1,2,3. This solution is then written as boldu=Aβ−αboldF with boldF=A−βboldf with β positive integer. The approximate solution method we propose and study is based on the best uniform rational approximation of the function tβ−α for 0

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Pore-scale modeling and simulation of reactive flow in porous media has a range of diverse applications, and poses a number of research challenges. It is known that the morphology of a porous medium has significant influence on the local flow rate, which can have a substantial impact on the rate of chemical reactions. While there are a large number of papers and software tools dedicated to simulating either fluid flow in 3D computerized tomography (CT) images or reactive flow using pore-network models, little attention to date has been focused on the pore-scale simulation of sorptive transport in 3D CT images, which is the specific focus of this paper. Here we first present an algorithm for the simulation of such reactive flows directly on images, which is implemented in a sophisticated software package. We then use this software to present numerical results in two resolved geometries, illustrating the importance of pore-scale simulation and the flexibility of our software package.

In this paper we consider efficient algorithms for solving the algebraic equation A α u = f , 0 < α < 1, where A is a properly scaled symmetric and positive definite matrix obtained from finite difference or finite element approximations of second order elliptic problems in R d , d = 1, 2, 3. This solution is then written as u = A β−α F with F = A −β f with β positive integer. The approximate solution method we propose and study is based on the best uniform rational approximation of the function t β−α for 0 < t ≤ 1, and the assumption that one has at hand an efficient method (e.g. multigrid, multilevel, or other fast algorithm) for solving equations like (A + cI)u = F, c ≥ 0. The provided numerical experiments confirm the efficiency of the proposed algorithms.

Novel parallel algorithms for the solution of large FEM linear systems arising from second order elliptic partial differential equations in 3D are presented. The problem is discretized by rotated trilinear nonconforming Rannacher-Turek finite elements. The resulting symmetric positive definite system of equations Ax = f is solved by the preconditioned conjugate gradient algorithm. The preconditioners employed are obtained by the modified incomplete Cholesky factorization MIC(0) of two kinds of auxiliary matrices B that both are constructed as locally optimal approximations of A in the class of M-matrices. Uniform estimates for the condition number κ(B −1 A) are derived. Two parallel algorithms based on the different block structures of the related matrices B are studied. The numerical tests confirm theory in that the algorithm scales as O(N 7/6 ) in the matrix order N .

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We simulate the thermal and electrical processes, involved in the radio-frequency ablation procedure. In this study, we take into account the observed fact, that the electrical conductivity of the hepatic tissue varies during the procedure. With the increase of the tissue temperature to a certain level, a sudden drop of the electrical conductivity is observed. This variation was neglected in some previous studies.The mathematical model consists of two parts -electrical and thermal. The energy from the applied AC voltage is determined first, by solving the Laplace equation to find the potential distribution. After that, the electric field intensity and the current density are directly calculated. Finally, the heat transfer equation is solved to determine the temperature distribution. Heat loss due to blood perfusion is also accounted for.The simulations were performed on the IBM Blue Gene/P massively parallel computer.