\documentstyle{article} \begin{document} %\markboth {R.P.Fedorenko}{ FINITE SUPRELEMENTS METHOD.} \begin{center} {\Large FINITE SUPRELEMENTS METHOD.}\\[1mm] R.P.~Fedorenko \end{center} \begin{center} \section{The main purposes and the main idea } \section*{of the method} \end{center} The Finite SuperElements Method (FSEM) is a special kind of FEM. It was proposed more then twenty years ego and now we have a numbers of its successful applications in different difficult computational problems. The first and the main purpose of FSEM was and remains the mathematical modeling of the nuclear reactors, bat there are numerous another possible applications. Our first publication on FSEM was \cite{b1} (1976). Just the same time Briggs L.L. and Lewis E.E. \cite{b2} begun to develop similar ideas for the same purpose; they called their approach The Method of Supercells. Let us regard the simplest problem where FSEM may by used and its basic construction may by explained without technical difficulties. Let the 1-dim. problem is to be solved \begin{equation} \label{f1} \frac{d}{dx} D(x) \frac{du}{dx} - A(x)\,u = f(x); \qquad x \in [0,\,N\times H] \end{equation} with any boundary conditions. Let the domain of definition of the function, $u(x)$, consists of N intervals of length H and each elementary interval has a composed inner structure. It means that coefficients $D(x), A(x)$ are composed functions over interval $H$. Let we are forced to use a grid with step $H$ and $H$ is "optically large" interval (it means $AH^2/D \approx 10$). And lastly let there are a few different types of such intervals and domain of solution is any combination of such intervals. Similar situation is typical in mathematical modeling of nuclear reactors and composite materials (in 2- or 3-dim. case of course). FSEM is a computational techniques which (under certain conditions of course) allows to construct the difference scheme taking into account the composed structure of the elementary grid cell. An approximate solution is sought as grid function $U_n, n=0,1,...,N$ , defined in grid nodes $nH$ . The first step is constructing of SE. Super Element is elementary interval $0 \le x \le H$ . The coefficients $D(x),A(x), x\in [0,\,H]$ and some special functions $q_i(x),i=1,..,I$ are defined for each type of elementary cell which may be met in the problem to be solved. These functions must be marked by special index, number of SE type. SE must be equipped with special basis formed by functions $\varphi_l ,\; l=1,2$, and $ \varphi_{0,i}, \; i=1,..,I$. These functions are defined by solution of equation (1) on interval $[0;H]$ with special boundary conditions and right-hand terms: $$ for\; \varphi_1 : \;\varphi(0)=1,\; \varphi(H)=0, \;f(x)=0 $$ $$ for \;\varphi_2 : \;\varphi(0)=0, \;\varphi(H)=1, \;f(x)=0 $$ $$ for \;\varphi_{0,i} : \;\varphi(0)=0, \;\varphi(H)=0, \;f(x)=q_i(x), \;i=1,...I $$ It is supposed that "real" function $f(x)$ over each H-interval may be approximated by linear combination of functions $q_i(x)$: $$ f(H+s) \approx \sum_{i=1}^{I} Q_{i} q_i(s), \quad s \in(0,H). $$ In common case one calculate the basis function by standard FDM using grid with step $h \ll H$ . An approximate solution $U(x)$ is represented in form \begin{equation} \label{f2} U(x)=U_n\varphi_1^{t}(x) + U_{n+1} \varphi_2^{t}(x) + \sum_{i=1}^{i=I(t)} Q^{n+1/2}_i\,\varphi_{0,i}^t (x); \qquad x \in [nH,(n+1)H] \end{equation} where t (more accurate it must be denoted $t(n+1/2)$) is a type index of SE placed on interval $[\,nH,\,(n+1)H \,]~$, $Q^{n+1/2}_i~$ are coefficients of approximation of function,$f(x)$ for this interval. For arbitrary values $U_n$ function $U(x)$ (\ref{f2}) is "piece-wise exact solution" (here and bellow we ignore the errors of calculation of the basis functions and approximation of the function,$f(x)$). To be an exact solution $U(x)$ must have continue "flow" $D\frac{dU}{dx}$ in the points $nH,\;n=1,...,N-1$. The conditions of the continuity of the flow form the difference scheme. To construct it we must not store basis functions, but only some functionals of them. These functionals are following ``flows'': %\begin{equation} \begin{eqnarray} \Pi^{t,\lambda}_l =D \frac{d \varphi^t_l}{d\nu} (x_\lambda ), \quad l=1,2; \; \; \lambda=1,2 \nonumber \\ \label{f3} \\ \Pi^{t,\lambda}_{0,i} = D \frac{d \varphi^t_{0,i}}{d\nu} (x_\lambda ), \quad \l=1,2; \; \; i=1,...I(t) \nonumber \end{eqnarray} %\end{equation} where $t$ is a type of SE, $x_1=0,\;x_2=H$, $\nu$ is direction of outer normal to bound of SE. Using these functionals one can to form the flow continuity condition in point $nH$: %\begin{equation} \begin{eqnarray} U_{n-1}\,\Pi^{t(n-1/2),2}_1 + U_n\,(\Pi^{t(n-1/2),2}_2 + \Pi^{t(n+1/2),1}_1 \,) + U_{n+1} \, \Pi^{t(n+1/2),1}_2 + \nonumber \\ \label{f4} \\ \sum_i Q_i^{n-1/2} \,\Pi^{t(n-1/2),2}_{0,i} + \sum_i Q_i^{n+1/2} \,\Pi^{t(n+1/2),1}_{0,i}=0 \nonumber \end{eqnarray} %\end{equation} This equation looks like a standard 3-point difference equation. It may be obtained by standard techniques after suitable choice of "homogeneous" piece-wise constant coefficients D,A. Note that these homogeneous values depend on the type of SE themselves as well as on difference scheme used and on the types of neighboring cells. So standard homogenization of each SE separately from another and calculation of homogeneous coefficients $D,A$ independently of difference approximation is not suitable from our point of view. The equation (\ref{f2}) is so called "exact scheme". Unfortunately, such a schemes exist only in 1-dim. case and are not very interesting. \vspace{1cm} \section{Superelements in 2-dim. and 3-dim.} \section*{~~~~~~~~~~~~~~~~ problems.} Let us restrict our consideration to 2-dim. case. Generalization on 3-dim. problems is evident but clumsily. Let the problem to be solved is 2-group diffusion equation in rectangular domain $NH\times NH$ \begin{eqnarray} %\begin{equation} div\,D_1grad\,u_1 \;-\;A_{11}\,u_1 \; + \; A_{12} \,u_2 = 0 \nonumber \\ \label{f5} \\ div\,D_2grad\,u_2 \;-\;A_{22}\,u_2 \; + \; A_{21} \, u_1 = 0 \nonumber %\end{equation} \end{eqnarray} Let the domain consists of $N\times N$ elementary cells $H\times H$ and each cell has a composed structure similar to the one shown on fig.1 (the coefficients of the equation have essentially different values in different parts of such an elementary cell). \begin{picture}(400,220) \put(157,2){Fig. 1} \put(10,30){\line(1,0){120}} \put(10,30){\line(0,1){120}} \put(130,30){\line(0,1){120}} \put(10,150){\line(1,0){120}} \put(70,90){\circle{25}} \put(70,90){\circle*{15}} \put(70,90){\circle{30}} \put(10,30){\circle*{5}} \put(10,150){\circle*{5}} \put(130,30){\circle*{5}} \put(130,150){\circle*{5}} \put(16,36){1} \put(16,155){4} \put(120,155){3} \put(120,35){2} \put(8,90){\line(1,0){4}} \put(2,88){A} \put(70,148){\line(0,1){4}} \put(70,154){B} \put(128,90){\line(1,0){4}} \put(135,88){C} \put(70,28){\line(0,1){4}} \put(70,21){D} \put(180,30){\line(1,0){120}} \put(180,30){\line(0,1){120}} \put(300,30){\line(0,1){120}} \put(180,150){\line(1,0){120}} \put(240,30){\line(0,1){120}} \put(180,90){\line(1,0){120}} \put(180,30){\circle*{5}} \put(240,30){\circle*{5}} \put(300,30){\circle*{5}} \put(180,90){\circle*{5}} \put(240,90){\circle*{5}} \put(300,90){\circle*{5}} \put(180,150){\circle*{5}} \put(240,150){\circle*{5}} \put(300,150){\circle*{5}} \put(210,60){t1} \put(270,60){t2} \put(210,120){t4} \put(270,120){t3} \put(175,20){k-1} \put(238,20){k} \put(295,20){k+1} \put(160,30){m-1} \put(165,90){m} \put(160,150){m+1} \put(238,88){\line(-1,0){28}} \put(238,92){\line(-1,0){28}} \put(238,88){\line(0,-1){28}} \put(242,88){\line(0,-1){28}} \put(238,92){\line(0,1){28}} \put(242,92){\line(0,1){28}} \put(242,92){\line(1,0){28}} \put(242,88){\line(1,0){28}} \put(245,95){\large $\omega_{k,m}$} \end{picture} Let we intend (or are forced) to use the grid with step size H, so approximate solution will be presented by grid function, $U_{k,m}$ , defined in nodes $(kH,mH)$. We suppose of course that the number of different possible types of such cells is essentially less than number of cells $N^2$. Let us regard each cell as SE. The basis in SE consists of 8 two-dimensional functions $\varphi^{\alpha}_{l,\beta}(x,y)$. Here $\alpha =1,2$ is index of component of vector function,$\;\varphi$; an index of basis function $l,\,\beta$ consists of number of node,$\; l=1,2,3,4$, and number of component $\,\beta=1,2$. These basis functions are the solutions of the equation (\ref{f5}) in square $H\times H$ with special boundary values. They are defined by following procedure. For any $l, \beta$ at first one define the values in nodes of SE: $$ \varphi^{\alpha}_{l,\beta}(x_n,y_n) = \delta^{\alpha}_{\beta} \times \delta^{l}_{n} ,\; n=1,2,3,4;\; \alpha =1,2. $$ Then the boundary values are interpolated piece wise linearly from nodes of SE on its sides. And finally the basis function is prolonged into SE by solution of the equation (\ref{f5}) taking into account the inner structure of SE; any suitable standard FDM with step $h \ll H$ may be used. Using this basis each 2-dim. grid function $U^1_{k,m},U^2_{k,m}$ may be convert into continue vector function $U(x,y)$, and this function is continue "piece wise exact solution" of the initial problem. To be exact solution $U(x,y)$ must have continue "flows" $(D\,grad\,U,\nu)$ on the grid lines. By special choice of the values $U_{k,m}$ one can provide only any weak continuity of the flows. Namely the conditions of weak continuity of the flows form the difference equations of FSEM. To construct them it is necessary to calculate and store some functionals of basis functions. Let the bound of SE is divided into four parts $\sigma_{\lambda},\, \lambda=1,..,4$; each part is associated with one of the nodes of SE. For example, $\sigma_1$ is D1A (see fig.1). The functionals to be calculated and stored are following "flows": \begin{equation} \label{f6} \Pi_{l,\beta}^{t, \alpha , \lambda}=\int_{\large \sigma_{\lambda}} D_{\alpha} \frac{\partial\varphi^{t,\alpha}_{l,\beta}}{\partial \nu} ds \end{equation} where $\nu$ is outer normal to the bound of SE, $t$ is a type of SE. To explain the construction of the difference equation in grid point $[k,m]$ let us consider fig.1~. The point $[k,m]$ is surrounded by four SE, $t_1,t_2,t_3,t_4$ are their types. It is connected with special region $\omega_{k,m}$, two-side cross (see fig.1). The condition of the weak continuity of the flow is used in the form: \begin{equation} \label{f7} \int_{\large \partial \omega_{k,m}} D_{\alpha} \frac{\partial U^{\alpha}} {\partial \nu} ds = 0, \quad \alpha=1,2. \end{equation} The contour $\partial \omega_{k,m}$ consists of four parts disposed in different cells; corresponding part of integral (\ref{f7}) is a sum of the flows multiplied by values of $U_{k',m'}$ in nodes of this cell. Omitting simple but bulky calculation we obtain the difference equation \begin{equation} \label{f8} \sum_{\beta=1}^{2}\,\sum_{k'=-1}^{1}\,\sum_{m'=-1}^{1}\, C^{\alpha,k',m',\beta}_{k,m} \, U^{\beta}_{k+k',m+m'} = 0, \,\,\alpha=1,2. \end{equation} The calculation of the scheme coefficients, C, is similar to the well known assembling of stiffness matrix in FEM: $$ C^{1,-1,-1,1}_{k,m}= \Pi^{t1,1,3}_{1,1};\quad C^{1,-1,-1,2}_{k,m}= \Pi^{t1,1,3}_{1,2};\quad C^{2,-1,-1,1}_{k,m}= \Pi^{t1,2,3}_{1,1}; $$ $$ C^{2,-1,-1,2}_{k,m}= \Pi^{t1,2,3}_{1,2};\quad C^{1,0,-1,1}_{k,m}= \Pi^{t1,1,3}_{2,1} + \Pi^{t2,1,4}_{1,1}; $$ $$ C^{1,0,-1,2}_{k,m}= \Pi^{t1,1,3}_{2,2} + \Pi^{t2,1,4}_{1,2};\quad\cdots . $$ So (\ref{f8}) is common 9-point (27-point in 3-dim problem) scheme and it is impossible to separate approximation of the term $divDgrad\,U$ from the approximation of term $A\,U$. Often for reconstruction of the solution inside of cell $H\times H$ and another applications it is necessary in addition to the functionals (\ref{f7}) to calculate and store some special functionals. {\it Superelements of second order.} More accurate scheme may be obtained using SE with additional nodes in points $A,B,C,D$, see fig.1. The bound of SE is divided into 8 equal parts with centers in nodes; just the same functionals (\ref{f6}) must be calculated and stored for $\alpha , \beta \in[1,2];\;l,\lambda \in [1,8]$. The basis on each side of SE is formed by functions $s/H,\; (H-s)/H,\; s(H-s)/(H^2/4)$. An approximate solution is presented by grid function $U_{k,m},\;U_{k+1/2,m},\;U_{k,m+1/2}$. And the weak continuity of the flow similar to (\ref{f7}) produces 13-point scheme in points $[k+1/2,m]$ and $[k,m+1/2]$, and 21-point scheme in points $[k,m]$ (52 and 88 coefficients C respectively). So FSEM unites some typical aspects of FDM (regular grid with standard geometric form of cells), FEM (the basis with local support) and Boundary Integral Equations Method. Note that basis functions used in FSEM form the standard interpolating basis (decomposition of the unit) only on surface of SE. Using FSEM of low order 1-2 one obtain the equations of type (\ref{f8}) which looks like the equations of FDM. More and more increasing the order of SE one obtain the equations more and more similar to the boundary integral ones. The functionals (\ref{f6}) may be regarded as any rough approximation of the Poincare-Steclov operator (it maps the boundary values of arbitrary solution of equation of type (\ref{f5}) into the boundary values of its normal derivatives). In principle this approach allows to exclude any part of computational region, replacing it by non-local boundary conditions. The effective application of FSEM is based on following main supposition: {\it The step size H is big with respect to the smoothness of the solution in usual sense, but it is small with respect to the smoothness of its restriction on grid lines.} It means that solution, $U(x,y)$ may be approximated with needed accuracy by piece wise linear function {\it on grid lines}. But it can not be approximated inside of grid cells by any standard basis functions of FEM, say by piece wise bilinear functions. In other words the inner structure of the grid cells causes the strong local deformation of the solution $U(x,y)$. In mathematical models of nuclear reactors such deformations of the neutron fields are caused by fuel elements and absorbing rods. Note that standard study of the convergence under the condition $H \to 0$ has no any sense for FSEM. \section{ Some applications of FSEM.} The possibilities of FSEM was tested by solution of some special problems with rather composed solutions. It must be noted that in all cases the grids with big step size was used. This section contains a short survey of these tests. {\it 1. Two group 2-dim. diffusion equation.} The system (\ref{f5}) was solved in region $5H\times 5H$, composed from strongly multiplying and strongly absorbing cells $H \times H$, the grid step H was big ($AH^2/D \approx 10$). Accuracy was checked by comparison with "exact" solution (it was obtained by calculation on grid with step $h=H/12$. The numerical solutions were rather good< but not very accurate. The FSEM of second order was realized too, numerical results were very good. More details see \cite{b3}. {\it 2. The problems with holes.} In some problems of underground hydrodynamics it is necessary to take into account the influence of the holes. The computational problem is connected with the fact that the radius of the hole is for example 100 times less than admissible step of the grid H. We have used FSEM in calculation of the state flow in the underground crack caused by influence of two holes. The size of grid step was 17 m., the radius of hole 10 cm. One of the equations of the problem was the equation in pressure, P $$ div D(x,y)grad P + Q(x,y) = 0,\qquad (x,y) \in \Omega \setminus (\bigcup_{i=1}^{i=2} \omega_i\,) $$ where $\omega_i$ are small holes, on their bounds was posed "inner" boundary conditions $P=p_i$. There was another difficulty connected with the degeneration of the problem: $D(x,y) \to 0$, when $ (x,y) \to \partial \Omega$; so classical boundary condition on $\partial \Omega$ were replaced by condition " boundedness of pressure, P on $\partial \Omega$" . But this question is outside of the subject of this communication. The effectiveness of FSEM was checked by use of the grid with step $H/2$ \cite{b4} and by solving of specially constructed problems with known solutions close to the solution to be find \cite{b5},\cite{b6}. {\it Diffusion in multi connected region.} In \cite{b7} the equation $\Delta U=0$ in region $\Omega \setminus (\cup_{i=1}^{i=I} \omega_i)$, where $\Omega$ is rectangle $X \times Y \times Z$, ~ $\omega_i$ are low cylinders with radius $R$ and height $h$ (model of the crack, $R=11$, $h=1.5$). The number of the cracks I was rather big (say 25 -100). In this problem SE was rectangle $50 \times 50 \times 5$. There were two types of SE: homogeneous and with crack,. The region $\Omega$ was an assemble $10 \times 10 \times 10$ of such SE. On outer bound $\partial \Omega$ function U is given. On inner bounds $\partial \omega_i$ the inner boundary conditions are posed $U=0$. In one of the problems solved there was a block of $5 \times 5$ cracks in plane $z=Z/2$. Respectively computational grid consisted of $10 \times 10 \times 10$ cells, $5 \times 5$ of them were SE with crack. Another solution of this problem was obtained with using SE of size $50/\sqrt{2} \times 50/\sqrt{2} \times 5$ with the same crack in center. In plane $xy$ this grid was rotated $\pi /4$ with respect the first one, the cracks were situated in 25 `black' cells. And third solution was obtained using SE of size $50 \times 50 \times 5$ of second order. It had 9 nodes on top side and 9 nodes on bottom one, basis consisted of 18 functions, numerical solution was described by grid function $U_{i,j,k}, \; U_{i+1/2,j,k}, \; U_{i,j+1/2,k}, \; U_{i+1/2,j+1/2,k}$. The most interesting results were the flows into the cracks. For example the flow in central crack was 63, 60 and 50 in different approximate solutions. (the last value is believed to be the more accurate). More details see \cite{b7}, \cite{b8}. {\it FSEM in special problem of elasticity theory.} In \cite{b9} is described an application of FSEM to numerical solution of following problem: in cylindrical region $\Omega$ the Lame equation in 3-vector of displacements, $U(x,y,z)$ is to be solved $$ div \,T(U)=0 \, , \qquad \mbox{where $T(U)$ is stress tensor.} $$ The body ($\Omega$) consists of 30 stiff vertical rods separated by soft material, their elasticity modules differ 100 times. The grid in plane $xy$ consisted of 30 hexagonal cells. SE in this case was hexagonal cylinder (radius 10, height 10) with vertical cylindrical rod (radius 5) in the center, it has 12 nodes. The basis in SE consists of 36 functions, the symmetry of SE allows to calculate all $36 \times 36$ functionals of type (\ref{f6}) after solution only 3 special boundary value problems in SE. But definition of the boundary values of the basis functions is a special problem, which has no unique natural solution. The cause is the rod crossing top and bottom sides of SE. In \cite{b9} the basis was constructed by following procedure. An hexagonal cylinder of double height 20 was used. At first the values of displacement $U$ is assumed to be 0 in all 18 nodes of this double SE. Then in the only one node in the middle of vertical edge only one component of displacement $U$ is assumed to be 1. Then defined in nodes values $U$ by standard bilinear interpolation generate the function $U$ on the surface of double SE. After all functionals (\ref{f6}) are calculated (only a half of the double SE is used of course) the difference scheme is constructed in the usual fashion. It is common 39-point scheme with 351 coefficients. The detailed qualitative analysis of numerical results obtained is made in \cite{b9}. Unfortunately we have no another tools to check them. \section{ Some problems and perspectives.} From preceding it is clear that schemes obtained according FSEM are rather cumbersome. Generalization its construction for M-group system of type (\ref{f5}) gives the 9-point scheme with $9 \times M^2$ coefficients C. Equipment of SE with basis need $4 \times M$ boundary value problem for system of M elliptic equations to be solved (in 3-dim. case these numbers are $27 \times M^2$, $8 \times M$ respectively). For SE of second (and higher) order these numbers become too large. In mathematical modeling of nuclear reactors which deals with multi group systems of type (\ref{5}) it is a practice to construct the computational schemes for equation for only one group with source: $$ \label{f10} div \,D_m \,grad \,U^m \;-\sigma_m U^m + S_m, \qquad m=1,...,M. $$ Interaction between different components of U is realized by `outer' iterations with recalculation of source, $ S_m(U^1,...U^{m-1},U^{m+1},...U^M)$. This approach is very attractive because of independence of the main code on group number,M In such a way are constructed the most of so called `nodal schemes'. We begun the work described in \ref{b3} from attempt to use this simplified form of FSEM. The results were negative. Here we shall try to explain the causes of the possible defect of such approach using simple 1-dim. equation (\ref{f5}). In that case FSEM gives us exact solution of the equations, obtained from (\ref{f5}) after replacement of the terms $A_{1,2}u_2(x),\;A_{2,1}u_1(x)$ by piece wise constant functions $$ (1/H) \int_{nH}^{(n+1)H} A_{1,2}u_2 (x)\,dx\, , x \in (nH,(n+1)H) $$ To obtain such a solution it is necessary in addition to the flows (ref{f3}) to calculate and store the mean values of the basis functions,$\varphi(x)$. Note that step of difference scheme is rather big ($A_{m,m} \,H^2/D_m \approx 10-20,\;m=1,2$) and we use only one function $q_{0,1}(x)=1$. The values $D_m,A_{m,n},\,H=14.7$ are taken from rather realistic mathematical model of reactor VVER-440, \cite{b11}. The equation (\ref{f11}) with constant (homogeneous) coefficients and source, S has evident exact solution (the constructions of nodal schemes are essentially based on this solution ); $$ \label{f12} U(x)=ae^{\lambda x} = be^{-\lambda x} -S/\sigma, \quad \mbox{where} \lambda=\sqrt{ \sigma/D} $$ Note that functions $e^{\pm \lambda x}$ change essentially over grid cell $0 < x< H$. But exact solution of the full system (\ref{f5}) consists of another exponents $e^{\mu x}$, where $\mu$ are kernels of characteristic equation $$ \mu ^4 \,- 2a \mu ^2\,+ d=0,\mbox{ where} a =(A_{1,1}/D_1 + A_{2,2}/D_2)/2, \; d=(A_{1,1}A_{2,2}-A_{1,2}A_{2,1})/(D_1 D_2) $$ In most SE $d \ll A_{1,1}A_{2,2}$ (except SE corresponding to rods, where $A_{1,2}=0$ ). So this equation has following kernels: $$ \mu_{1,2} \approx \pm \sqrt{2a}, \quad \mid \mu H \mid \gg 1 ; \qquad \mu_{3,4} \approx \pm \sqrt{d/2a},\quad \mid \mu H \mid \ll 1 $$ The first pair of kernels defines the boundary layers, the second one defines linear functions ($e^{\mu x} + e^{-\mu x} \approx 2, (e^{\mu x} - e^{-\mu x})/\mu \approx 2x$). The basis of simplified FSEM contains the functions describing boundary layers and function $\varphi_{0,1}(x) \equiv 1$, so function $x$ is lost. It seems to be the main cause of the error of numerical solution. Distinction of $\mu_{1,2}$ from $\pm \sqrt{A_{i,i}/D_i}$ seems us unessential. If so, an improving of the method may be obtained by increasing of the functions for approximation of the source, $S$. In the table 1 are presented approximate and exact functions $u_2(nH),\;n=0,15$ (exact solution is found by the same method with step $h=H/20$) % \begin{eqnarray} ~\\ Table 1\\ \begin{tabular}{l} % -------------------------Table 1 ------------------------------------- \\ \hline 0~~18~~23~~27~~21~~~~8~~-12~~-32~~-53~~-70~~-65~~-52~~-25~~20~~69~100 \\ 0~~14~~17~~18~~10~~~-3~~-17~~-29~~-38~~-45~~-38~~-22~~~08~~46~~81~100 \\ \hline % ----------------------------------------------------------------------- \end{tabular} ~\\ % \end{eqnarray} {\it FSEM and local refinement of the grid and parallel processes.} In mathematical model of nuclear reactors often in plane $xy$ are using hexagonal grids. For example in \cite{b11} there are 421 hexagonal cells, so cells of 3-dim. grid are hexagonal cylinders, their centers are grid points. The coefficients of multi group diffusion equations depends on `state' of the cell (it consists of temperatures of fuel and water, concentrations of xenon, samarium and so on). In fact the cell is an assembly of about 300 thin vertical rods (fuel elements,FE) and homogeneous coefficients are defined namely for fuel element because its state may be described by seven, for example, parameters. A special interpolation is used; it is based on the data calculated before. Now we are using relatively rough model, where the states of all FE in assembly is supposed to be equal. It is actual problem to construct more accurate models taking into account the individual state of each FE and its change over time (in problems of reactor dynamics). So state of the assembly is described by `function' (or by vector of very large dimension). It is impossible to use any interpolation for calculation of the homogeneous coefficients for assembly. To use the grid with one grid point per each FE and to solve all equations as unique system is too expensive. But it is necessary to calculate the consequences of any emergency of only FE. FSEM offers some possibilities in this problem. Each hexagonal cylindrical cell (big cell which in \cite{b11} is an elementary grid cell) is regarded as separate SE, from computational point of view similar to full reactor in \cite{b11}. In such SE must be solved separate tasks, boundary value problems for basis functions. Each problem can be solved independently from another by separate processor. After all problems for all big cells are solved the connecting global system of equations for full reactor may be formed. It will be similar to the difference equations in \cite{b8}, \cite{b9} on rough grid used in \cite{b11} for example, its solution is relatively easy. The only but essential difference from scheme used in \cite{b11} is connected with positions of the grid points: in centers of cells in \cite{b11} and in their vertexes in \cite{b8}. And finally this `rough' solution is interpolated on the bounds of beg cells, it will be necessary to solve separate boundary value problems for all big cells. \begin{thebibliography}{99} \bibitem{b1} L.G.Strakhovskaja, R.P.Fedorenko. On one special difference scheme. Num. Math. of continue medium mech., Novosibirsk, CC SO AS USSR, v.7, N4, p.149-163 (in russian) \bibitem{b2} Briggs L.L., Lewis E.E. A two-dimensional constrained finite element method for nonuniform lattice problem. Nucl.Sci.Engn. 1980, v.75, p.76-87. \bibitem{b3} L.G.Strakhovskaja, R.P.Fedorenko, The finite elements method for two-group diffusion equation. Preprint KIAM, AS USSR, N31, 1977 (in russian) \bibitem{b4} Zazovskii A.F., Lemkha A.V., Fedorenko R.P. Numerical analysis of elastic-hydrodynamic problem on circulation of the liquid in underground crack. KIAM AS USSR, Preprint N 1, 1987 (in russian) \bibitem{b5} Zazovskii A.F., Lemkha A.V., Fedorenko R.P. Nonlinear effects of the circulation of the liquid in underground crack. KIAM AS USSR, Preprint N 219, 1987 (in russian) \bibitem{b6} L.G.Strakhovskaja, R.P.Fedorenko, On one variant of Finite Elements Method. J. of Comp. Math. and Math.Phys. 1979, v.19, N 4, p.950-960 \bibitem{b7} L.G.Strakhovskaja, R.P.Fedorenko, The calculation of diffusion in multi connected region by Finite Superelements Method. Preprint KIAM, AS USSR, N171, 1987 (in russian) \bibitem{b8} Fedorenko R.P. Finite superelements method and multigrid method in problems of elasticity theory. CFD Journal, vol.5, N 2, 1996 \bibitem{b9} L.G.Strakhovskaja, R.P.Fedorenko, The calculation of the stress state of 3-dim. composed body by Finite SuperElements Method. Preprint KIAM, AS USSR, N67, 1994 (in russian) \bibitem{b11} M.Telbisz, A.Kereszturi. Results of a three-dimensional hexagonal kinetic benchmark problem. Proceeding of 2-nd symposium of Atomic Energy Research (AER), Paks (Hungary,1992 \end{thebibliography} \end{document} %\end{document}