ORIGINAL_ARTICLEThe paper is devoted to the parallel solution of the two-dimensional stationary reaction-diffusion problem. By the usage of parallel approach to the linear algebra representa- tion we create the parallel algorithm for computing a numerical solution of the two-dimensional stationary reaction-diffusion problem. We compare calculation times of computing the approximate solution of the system of (linear) difference equations for different sizes of the system matrix by the numerical conjugate gradient method on 1, 2, 3, and 4 processors, respectively.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_01.pdf2012-12-21814Stationary reaction – diffusion problemparallel linear algebrafinite difference methodconjugate gradient methodDanielaBímová1AUTHOR

ORIGINAL_ARTICLEIn the paper, an algorithm for finding an optimal or almost optimal permutation for anordering of elements of a matrix, which is sparse, asymmetric and reducible, is suggested.Using this algorithm we can solve large sparse systems of linear equations more efficiently.The algorithm is a modification of the algorithm presented in [6], therefore the sameindications and symbols are use.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_02.pdf2012-12-211521Asymmetric reducible sparse matrixalgorithmDanielaBittnerová1AUTHOR

ORIGINAL_ARTICLEIterative methods for solving variational inequalities in infinite dimensional Hilbert spaces asa rule require some discretization. This leads to variational inequalities over families of spaces.In the present paper this problem is addressed by an iterative method with only a finite numberof steps at each discretization level. First, abstract methods are studied and later an optimalcontrol problem with elliptic state equations and some bound on the controls is considered. Thediscretization technique rests upon a nested family of piecewise linear C0-elements conformingfinite element discretizations.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_03.pdf2012-12-212231Optimal controliteration-discretizationprojection algorithmelliptic state equation finite element discretizationAndrzejCegielski1AUTHORChristianGrossmann2AUTHOR

ORIGINAL_ARTICLEOne of the most important part of adaptive wavelet methods is an efficient approximate multi- plication of stiffness matrices with vectors in wavelet coordinates. Although there are known algorithms to perform it in linear complexity, the application of them is relatively time con- suming and its implementation is very difficult. Therefore, it is necessary to develop a well- conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in any column is bounded by a constant. Then, matrix-vector multiplication can be performed exactly with linear complexity. We present here a wavelet basis on the interval with respect to which both the mass and stiffness matrices corresponding to the one-dimensional Laplacian are sparse. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in tensor product wavelet basis is also sparse. Moreover, the constructed basis has an excellent condition number. In this contribution, we shortly review this construction and show several numerical tests.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_04.pdf2012-12-213239Convection-diffusion equationsplinewaveletadaptivityDanaČerná1AUTHORVáclavFiněk2AUTHOR

ORIGINAL_ARTICLEThe paper is concerned with theoretical and computational issues of a numerical resolution ofhttps://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_05.pdf2012-12-214045WaveletHermite cubic splinessparse representationsDanaČerná1AUTHORVáclavFiněk2AUTHORZdenaOndračková.3AUTHOR

ORIGINAL_ARTICLEFirst multiwavelets have appeared around the early 1990s. The basic idea behind multiwavelets is simple: to replace the single scaling function by the multiscaling function 4t to have some additional desired properties. It seems to be an interesting trade off because multiwavelets provide higher order approximation with shorter support than single scaling function. Moreover, it is possible to have both symmetric and orthogonal multiwavelets while this is not possible for single wavelets. In recent years, several simple constructions of wavelet bases based on Hermite cubic splines were proposed. In this contribution, we shortly review these constructions, use these wavelets to solve numerically differential equations, and compare their performance.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_06.pdf2012-12-214655WaveletHermite cubic splineselliptic differential equationsDanaČerná1AUTHORVáclavFiněk2AUTHORGertaPlačková3AUTHOR

ORIGINAL_ARTICLEWe deal with a numerical solution of nonlinear convection-diffusion problems with the aid of the discontinuous Galerkin finite element (DGFE) method. We propose a new hp-adaptation technique, which is based on a combination of a residuum-nonconformity estimator and a regularity indicator. The residuum-nonconformity estimator consists of two building blocks (the residuum error indicator and the value of the nonconformity). The estimator marks mesh elements for a refinement. The regularity indicator decides if the marked elements will be refined by h- or p-technique. The residuum-nonconformity estimator as well as the regularity indicator are easily computable quantities. Moreover, the same technique estimates an algebraic error arising from an iterative solution of the corresponding nonlinear algebraic system. The performance of the proposed hp-DGFE method is demonstrated by several numerical examples. https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_07.pdf2012-12-215667hp-discontinuous Galerkin finite element methodresiduum-nonconformity indicator regularity estimatoralgebraic errorVítDolejší1AUTHOR

ORIGINAL_ARTICLEWe consider nonlinear algebraic systems arising from numerical discretizations of nonlinear partial differential equations of diffusion type. In order to solve them, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. We propose adaptive stopping criteria for both these solvers, based on an a posteriori error estimate which distinguishes the different error components, namely the discretization, linearization, and algebraic ones. Our estimates give a guaranteed error upper bound and also a robust error lower bound. Numerical experiments for the nonlinear Laplace equation, nonconforming finite element discretization, Newton linearization, and conjugate gradients algebraic solver illustrate the theory. https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_08.pdf2012-12-216875Nonlinear algebraic systemadaptive linearizationadaptive algebraic solution stopping criteriona posteriori error estimateAlexandreErn1AUTHORMartinVohralík.2AUTHOR

ORIGINAL_ARTICLEa vector which we have recently proposed in [1, 2]. The theoretical advantages of our schemehttps://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_09.pdf2012-12-217684Interval datacomputation of variancetheoretical analysisVáclavFiněk1AUTHORCtiradMatonoha2AUTHOR

ORIGINAL_ARTICLESystems of so called two-sided (max,min)—linear inequalities with variables on both sides will be studied. Optimization problems, the objective function of which is equal to the maximum of a finite number of continuous functions of one variable are considered. The set of feasible solutions in described by a system of two-sided (max,min)—linear inequalities with variables on both sides. A finite algorithm for finding the optimal solution of the problem is proposed. Keywords: Two-sided (max,min)—linear inequalities system; lower and upper bounds; max-min optimization problems. https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_10.pdf2012-12-218594Two-sided (maxmin)−linear inequalities systemlower and upper boundsmaxmin optimization problemsMahmoudGad1AUTHOR

ORIGINAL_ARTICLEWe solve the thermal dimensioning problem of the deep geological spent nuclear fuel repository, which means to estimate the maximum temperature in the repository caused by the heat generation of the spent fuel. We use a combination of the boundary element method for the exterior problem of heat discharge to the infinity and finite element method for a near- field thermal problem with a boundary condition expressed by the far-field problem solution. This combination is implemented within the simulation software ANSYS as the “far-field element”. The far-field element solution confirmed to well represent the heat discharge, in comparison with the variant of a standard FEM far-field problem solution and conventional boundary conditions (constant temperature or zero heat flow).https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_11.pdf2012-12-2195101heat conductionnumerical simulationfar-field elementmultiscaleANSYSMilanHokr1AUTHORJosefNovák2AUTHOR

ORIGINAL_ARTICLEIn this paper we deal with the development of a numerical method for the solution of the mo- dified equal width wave (MEW) equation – a very important equation with a cubic nonlinearity describing a large number of physical phenomena. The crucial idea of introduced approach is based on the discretization of the MEW equation with the aid of a combination of the discontin- uous Galerkin (DG) method for the space semi-discretization and the backward Euler method for the time discretization. The appended numerical experiments investigate the conservative properties of the MEW equation related to mass, momentum and energy, and illustrate the po- tency of this scheme, consequently.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_12.pdf2012-12-21102110Discontinuous Galerkinmethodmodified equal width wave equationsemi-implicit schemesolitary waveJiříHozman1AUTHOR

ORIGINAL_ARTICLEThis paper gives an overview of the main ingredients needed to incorporate reconstruction op- erators, as known from higher order finite volume (FV) and spectral volume (SV) schemes, into the discontinuous Galerkin (DG) method. Such an operator constructs higher order approxima- tions from the lower order DG scheme, increasing the order of convergence, while leading to a more efficient numerical scheme than the corresponding higher order DG scheme itself. We discuss theoretical, as well as implementational aspects and numerical experiments.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_13.pdf2012-12-21111119Higher order reconstructiondiscontinuous Galerkinfinite volumesVáclavKučera1AUTHOR

ORIGINAL_ARTICLEThe subject of the presented paper is a mathematical analysis and numerical solution of the sys- tem of nonlinear nonstationary reaction-diffusion equations. Firstly, using the invariant region technique, the proof of both the existence and uniqueness of the solution and problem data con- tinuous dependence is carried out. After time discretization of the problem the Galerkin finite elements method is applied and a priori error estimates of the method are derived. A suitable mesh adaptivity is discussed as well. The method is finally implemented and tested on several examples.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_14.pdf2012-12-21120128Invariant regionreactiondiffusionfinite elements methodadaptivityJanLamač1AUTHOR

ORIGINAL_ARTICLEA viscoelastic simply supported rotationally symmetric body, fixed on a base, is considered. The body is loaded by a flat plunger, which moves in the direction of the z axis by a constant velocity v. In this work the reaction force is computed. This allows us to compare numerical results with data from rheological experiment (see [6], [7]). The variational formulation of the problem is derived and transformed to cylindrical coordinates. Some results of numerical calculations are presented.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_15.pdf2012-12-21129137Viscoelasticityaxisymmetric hyperbolic problemsdimensional reductionPetrSalač1AUTHORIvoMatoušek2AUTHOR

ORIGINAL_ARTICLEStiffness matrix of the Dirichlet problem (auI)I = f with a homogeneous boundary value condition in a spline wavelet basis has O(n log n) non-zero elements [4]. We show that for a constant function a it is just O(n) and moreover we show that it can be stored in O(1) elements. This leads to a linear-time algorithm for multiplication by the wavelet matrix.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_16.pdf2012-12-21138146Dirichlet problemefficient implementationGalerkin methodspline waveletsMartinaŠimůnková1AUTHOR

ORIGINAL_ARTICLEWe consider an unsteady 1D singularly perturbed convection–diffusion problem. We discretize such a problem by the linear finite element method (FEM) on a Shishkin mesh and by a discon- tinuous Galerkin method in time. We present optimal a priori error estimates for low order time discretizations.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_17.pdf2012-12-21147154Convection–diffusionShishkin meshtime discontinuous Galerkin methodMiloslavVlasák1AUTHORHans–GöergRoos2AUTHOR

ORIGINAL_ARTICLEThe paper deals with the problem of non-parametric statistical modeling of 2-dimensional sur- faces from observed data, i.e. the regression analysis. In general, the model is constructed from a set of basal functions, as are the splines, gaussians and others. However, such model- ing means to estimate a large set of parameters (locations of functional units and parameters of their combination). We shall present two approaches allowing reduction of the number of needed parameters. Namely, a well known method of projection pursuit, and the less known method of Gordon surface. Further, we shall analyze possible serious consequences of sparse data to precision of model and uncertainty of prediction. Methods will be illustrated in artificial examples.https://acc-ern.tul.cz/archiv/PDF/ACC_2012_4_18.pdf2012-12-21155165Statisticsregression analysissplinesprojection pursuitGordon surfaceprediction errorPetrVolf1AUTHOR