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A curious property of oscillatory FEM solutions of one-dimensional convection-diffusion problems

Madden, Niall, Stynes, Martin (2012)

Applications of Mathematics 2012

Song, Yin and Zhang (Int. J. Numer. Anal. Model. 4: 127-140, 2007) discovered a remarkable property of oscillatory finite element solutions of one-dimensional convection-diffusion problems that leads to a novel numerical method for the solution of such problems. In the present paper this property is described using several figures, then a simple proof of the phenomenon is given which is much more intuitive than the technical analysis of Song et al.

A direct solver for finite element matrices requiring O ( N log N ) memory places

Vejchodský, Tomáš (2013)

Applications of Mathematics 2013

We present a method that in certain sense stores the inverse of the stiffness matrix in O ( N log N ) memory places, where N is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires O ( N 3 / 2 ) arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with O ( N log N ) operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular domains,...

A Language Engineering Architecture for Processing Informal Mathematical Discourse

Wolska, Magdalena (2008)

Towards Digital Mathematics Library. Birmingham, United Kingdom, July 27th, 2008

We present a modular architecture for processing informal mathematical language as found in textbooks and mathematical publications. We point at its properties relevant in addressing three aspects of informal mathematical discourse: (i) the interleaved symbolic and natural language, (ii) the linguistic, domain, and notational context, and (iii) the imprecision of the informal language. The objective in the modular approach is to enable parameterisation of the system with respect to the natural language...

A mesh free numerical method for the solution of an inverse heat problem

Azari, Hossein, Parzlivand, F., Zhang, Shuhua (2012)

Applications of Mathematics 2012

We combine the theory of radial basis functions with the finite difference method to solve the inverse heat problem, and use five standard radial basis functions in the method of the collocation. In addition, using the newly proposed numerical procedure, we also discuss some experimental numerical results.

A method to rigorously enclose eigenpairs of complex interval matrices

Castelli, Roberto, Lessard, Jean-Philippe (2013)

Applications of Mathematics 2013

In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices is proposed. Each eigenpair x = ( λ , ) is found by solving a nonlinear equation of the form f ( x ) = 0 via a contraction argument. The set-up of the method relies on the notion of r a d i i p o l y n o m i a l s , which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable.

A multilevel correction type of adaptive finite element method for Steklov eigenvalue problems

Lin, Qun, Xie, Hehu (2012)

Applications of Mathematics 2012

Adaptive finite element method based on multilevel correction scheme is proposed to solve Steklov eigenvalue problems. In this method, each adaptive step involves solving associated boundary value problems on the adaptive partitions and small scale eigenvalue problems on the coarsest partitions. Solving eigenvalue problem in the finest partition is not required. Hence the efficiency of solving Steklov eigenvalue problems can be improved to the similar efficiency of the adaptive finite element method...

A multi-space error estimation approach for meshfree methods

Rüter, Marcus, Chen, Jiun-Shyan (2015)

Application of Mathematics 2015

Error-controlled adaptive meshfree methods are presented for both global error measures, such as the energy norm, and goal-oriented error measures in terms of quantities of interest. The meshfree method chosen in this paper is the reproducing kernel particle method (RKPM), since it is based on a Galerkin scheme and therefore allows extensions of quality control approaches as already developed for the finite element method. Our approach of goal-oriented error estimation is based on the well-established...

A note on necessary and sufficient conditions for convergence of the finite element method

Kučera, Václav (2015)

Application of Mathematics 2015

In this short note, we present several ideas and observations concerning finite element convergence and the role of the maximum angle condition. Based on previous work, we formulate a hypothesis concerning a necessary condition for O ( h ) convergence and show a simple relation to classical problems in measure theory and differential geometry which could lead to new insights in the area.

A note on tension spline

Segeth, Karel (2015)

Application of Mathematics 2015

Spline theory is mainly grounded on two approaches: the algebraic one (where splines are understood as piecewise smooth functions) and the variational one (where splines are obtained via minimization of quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the well known tension spline (called also spline in tension or spline with tension). We present the results of...

A parallel method for population balance equations based on the method of characteristics

Li, Yu, Lin, Qun, Xie, Hehu (2013)

Applications of Mathematics 2013

In this paper, we present a parallel scheme to solve the population balance equations based on the method of characteristics and the finite element discretization. The application of the method of characteristics transform the higher dimensional population balance equation into a series of lower dimensional convection-diffusion-reaction equations which can be solved in a parallel way. Some numerical results are presented to show the accuracy and efficiency.

À propos du sens des textes mathématiques, un exemple : la théorie des «bifurcations dynamiques»

Claude Lobry (1992)

Annales de l'institut Fourier

L’interprétation dans la langue naturelle d’un résultat mathématique peut être délicate. La théorie des bifurcations fournit un exemple récent ou des problème d’interprétation ont motive des développements mathématiques nouveaux de la théorie. L’article plaide pour une plus grande prise en considération de ces questions dans les revues mathématiques.

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