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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.

a r t i c l e i n f o MSC: 65M60 Keywords: Vacuum freeze drying Zeolites Heat and mass transfer Parabolic PDE Finite element method MIC(0) preconditioning a b s t r a c tThe work is motivated by the problem of freeze-drying, which is a process of dehydrating frozen materials by sublimation under high vacuum. In particular, it concerns the mathematical modelling and computer simulation of the heat and mass transfer with the core in solving the time-dependent nonlinear partial differential equation of parabolic type.Instead of a uniform discretization of the considered time interval, an adaptive time-stepping procedure is applied in an effort to optimize the whole simulation. The procedure is based on the local comparison of the Crank-Nicolson and backward Euler approximations. The results of numerical experiments performed on a selected real-life problem are included.

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved.

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