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

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 $O\left(NlogN\right)$ memory places

Applications of Mathematics 2013

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

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

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

Applications of Mathematics 2013

In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices is proposed. Each eigenpair $x=\left(\lambda ,\right)$ is found by solving a nonlinear equation of the form $f\left(x\right)=0$ via a contraction argument. The set-up of the method relies on the notion of $radiipolynomials$, 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

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

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

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 $O\left(h\right)$ 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

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

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.

A short philosophical note on the origin of smoothed aggregations

Applications of Mathematics 2013

We derive the smoothed aggregation two-level method from the variational objective to minimize the final error after finishing the entire iteration. This contrasts to a standard variational two-level method, where the coarse-grid correction vector is chosen to minimize the error after coarse-grid correction procedure, which represents merely an intermediate stage of computing. Thus, we enforce the global minimization of the error. The method with smoothed prolongator is thus interpreted as a qualitatively...

A strengthening of the Poincaré recurrence theorem on MV-algebras

Applications of Mathematics 2012

The strong version of the Poincaré recurrence theorem states that for any probability space $\left(\Omega ,𝒮,P\right)$, any $P$-measure preserving transformation $T:\Omega \to \Omega$ and any $A\in 𝒮$ almost every point of $A$ returns to $A$ infinitely many times. In [8] (see also [4]) the theorem has been proved for MV-algebras of some type. The present paper contains a remarkable strengthening of the result stated in [8].

A sufficient condition for non-oscillatory behavior of some input/output models

Applications of Mathematics 2012

A simple condition sufficient for non-oscillatory behavior of input/output systems is formulated and discussed.

Adaptive finite element analysis based on perturbation arguments

Applications of Mathematics 2012

We review some numerical analysis of an adaptive finite element method (AFEM) for a class of elliptic partial differential equations based on a perturbation argument. This argument makes use of the relationship between the general problem and a model problem, whose adaptive finite element analysis is existing, from which we get the convergence and the complexity of adaptive finite element methods for a nonsymmetric boundary value problem, an eigenvalue problem, a nonlinear boundary value problem...

Algebraic classification of the Weyl tensor

Applications of Mathematics 2012

Alignment classification of tensors on Lorentzian manifolds of arbitrary dimension is summarized. This classification scheme is then applied to the case of the Weyl tensor and it is shown that in four dimensions it is equivalent to the well known Petrov classification. The approaches using Bel-Debever criteria and principal null directions of the superenergy tensor are also discussed.

Algebraic classification of the Weyl tensor: selected applications

Applications of Mathematics 2012

Selected applications of the algebraic classification of tensors on Lorentzian manifolds of arbitrary dimension are discussed. We clarify some aspects of the relationship between invariants of tensors and their algebraic class, discuss generalization of Newman-Penrose and Geroch-Held-Penrose formalisms to arbitrary dimension and study an application of the algebraic classification to the case of quadratic gravity.

An adaptive $hp$-discontinuous Galerkin approach for nonlinear convection-diffusion problems

Applications of Mathematics 2012

We deal with a numerical solution of nonlinear convection-diffusion equations with the aid of the discontinuous Galerkin method (DGM). We propose a new $hp$-adaptation technique, which is based on a combination of a residuum estimator and a regularity indicator. The residuum estimator as well as the regularity indicator are easily evaluated quantities without the necessity to solve any local problem and/or any reconstruction of the approximate solution. The performance of the proposed $hp$-DGM is demonstrated....

An international symposium honouring professor Karel Rektorys' eightieth. Foreword

Applications of Mathematics

Analytical solution of rotationally symmetric Stokes flow near corners

Applications of Mathematics 2013

We present analytical solution of the Stokes problem in rotationally symmetric domains. This is then used to find the asymptotic behaviour of the solution in the vicinity of corners, also for Navier-Stokes equations. We apply this to construct very precise numerical finite element solution.

Analytical solution of Stokes flow near corners and applications to numerical solution of Navier-Stokes equations with high precision

Applications of Mathematics 2012

We present analytical solution of the Stokes problem in 2D domains. This is then used to find the asymptotic behavior of the solution in the vicinity of corners, also for Navier-Stokes equations in 2D. We apply this to construct very precise numerical finite element solution.

Application of Richardson extrapolation with the Crank-Nicolson scheme for multi-dimensional advection

Applications of Mathematics 2013

Multi-dimensional advection terms are an important part of many large-scale mathematical models which arise in different fields of science and engineering. After applying some kind of splitting, these terms can be handled separately from the remaining part of the mathematical model under consideration. It is important to treat the multi-dimensional advection in a sufficiently accurate manner. It is shown in this paper that high order of accuracy can be achieved when the well-known Crank-Nicolson...

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