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The paper presents the method and instrumental system for modeling and optimizing technological modes of direct metal reduction processes in a jet-emulsion aggregate (JER). Stages of the method are considered. The first one is the problem statement: formation of target conditions, choice of the process type, the task and system of optimization criteria. The second stage includes selection of the object of study: setting parameters of input and output flows, process parameters, stages and subprocesses. The third one includes thermodynamic modeling to assess the final equilibrium state in which optimization problem is solved to determine the best conditions for implementation of the processes of metal reduction from oxides in model systems. The fourth stage is development of metallurgical technology (finding the optimum modes and ways for achieving these modes by specified output stream parameters). And the final one is process optimization in technical and economic indicators. As part of the fourth stage, the complex of mathematical models has been developed that reflects relationship of flows and processes in a metallurgical unit. The structure of instrumental system is presented, in which mathematical models and an algorithm for determining optimal technological modes are implemented. A set of optimization criteria has been developed and a scheme for solving two types of optimization problems are presented: finding optimal conditions for reduction processes in thermodynamic systems and determining optimal modes of direct metal reduction. Application of the method to develop optimal technological modes of direct metal production in a JER-type aggregate is shown: metal production from cast iron and mill scale; direct reduction of metal from dusty ores and iron-containing man-made materials; obtaining manganese alloys from carbonate and oxide ores; processing titanium-magnetite concentrates with an almost complete separation of iron-containing and titaniumcontaining component; and direct reduction of iron with associated production of high-calorie synthesis gas.

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Dur\'{a}n- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution., Comment: 18 pages, 2 figures, 3 tables

We consider an interior turning point problem with two exponential boundary layers. Its discretisation on a piecewise-uniform Shishkin mesh yields a scheme which is uniformly convergent (measured in the discrete maximum norm) of almost order one. Richardson extrapolation improves the accuracy to O ( N - 2 ln 2 N ) . Both can be proved under the assumption e ? C N - 1 .

The Il’in scheme is the most famous exponentially fitted finite difference scheme for singularly perturbed boundary value problems. In 1D, Kellogg and Tsan presented a precise error estimate for the scheme from which uniform first order convergence in the discrete maximum norm can be concluded. This estimate is optimal in the sense that one can supply easy examples with smooth data such that the order of uniform convergence of the Il’in scheme is one. In 2D the problem of proving an optimal error estimate remained open. Emel’janov conducted an error analysis giving uniform convergence orders close to one-half. Within the community of singularly perturbed problems it is a legend that this error estimate is sharp. Correcting this mistaken belief it is proven in the present paper that under some conditions the optimal uniform first order convergence of the Il’in scheme in 1D carries over to the two dimensional case. This new result is corroborated by numerical experiments which also shed light on the question in which cases the convergence rate deteriorates.

Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H1 seminorm leads to a balanced norm which reflects the layer behavior correctly. We prove error estimates in balanced norms and investigate also stability questions. Especially, we propose a new C0 interior penalty method with improved stability properties in comparison with the Galerkin FEM.

We analyze the convergence of a continuous interior penalty (CIP) method for a singularly perturbed fourth-order elliptic problem on a layer-adapted mesh. On this anisotropic mesh, we prove under reasonable assumptions uniform convergence of almost order k − 1 for finite elements of degree k ≥ 2. This result is of better order than the known robust result on standard meshes. A by-product of our analysis is an analytic lower bound for the penalty of the symmetric CIP method. Finally, our convergence result is verified numerically. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 838–861, 2014

The $$\varepsilon $$ ? -weighted energy norm is the natural norm for singularly perturbed convection-diffusion problems with exponential layers. But, this norm is too weak to recognise features of characteristic layers. We present an error analysis in a differently weighted energy norm--a balanced norm--that overcomes this drawback.

Recently, several classes of fourth order singularly perturbed problems were considered and uniform convergence in the associated energy norm as well as in a balanced norm was proved. In this proceedings paper we will extend some results by looking into L ∞ -bounds and postprocessing.

We present new results in the numerical analysis of singularly perturbed convection-diffusion-reaction problems that have appeared in the last five years. Mainly discussing layer-adapted meshes, we present also a survey on stabilization methods, adaptive methods, and on systems of singularly perturbed equations.

So far optimal error estimates on Bakhvalov-type meshes are only known for finite difference and finite element methods solving linear convection-diffusion problems in the one-dimensional case. We prove (almost) optimal error estimates for problems with exponential boundary layers in two dimensions.

Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on $$ N \times \sqrt N , \sqrt N \times N $$ and $$ \sqrt N \times \sqrt N $$ meshes. It is shown that the combination FEM yields (up to a factor ln N) the same order of accuracy in the associated energy norm as the Galerkin FEM on an N × N mesh, but it requires only $$ \mathcal{O} $$ (N 3/2) degrees of freedom compared with the $$ \mathcal{O} $$ (N 2) used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.