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Suitability of linearization of nonlinear problems not only in biology and medicine

Jana Vrbková (2009)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Biology and medicine are not the only fields that present problems unsolvable through a linear models approach. One way to overcome this obstacle is to use nonlinear methods, even though these are not as thoroughly explored. Another possibility is to linearize and transform the originally nonlinear task to make it accessible to linear methods. In this aricle I investigate an easy and quick criterion to verify suitability of linearization of nonlinear problems via Taylor series expansion so that...

Summation of slowly convergent series

Walter Gautschi (1994)

Banach Center Publications

Among the applications of orthogonal polynomials described briefly on my previous visit to this Center [9, §3.2] were slowly convergent series whose terms could be represented in terms of the Laplace transform at integer arguments. We proposed to sum such series by means of Gaussian quadrature rules applied to suitable integrals involving weight functions of Einstein and Fermi type (cf. [13]). In the meantime it transpired that the technique is applicable to a large class of numerical series and,...

Superapproximation of the partial derivatives in the space of linear triangular and bilinear quadrilateral finite elements

Dalík, Josef (2013)

Programs and Algorithms of Numerical Mathematics

A method for the second-order approximation of the values of partial derivatives of an arbitrary smooth function u = u ( x 1 , x 2 ) in the vertices of a conformal and nonobtuse regular triangulation 𝒯 h consisting of triangles and convex quadrilaterals is described and its accuracy is illustrated numerically. The method assumes that the interpolant Π h ( u ) in the finite element space of the linear triangular and bilinear quadrilateral finite elements from 𝒯 h is known only.

Superconvergence analysis and a posteriori error estimation of a Finite Element Method for an optimal control problem governed by integral equations

Ningning Yan (2009)

Applications of Mathematics

In this paper, we discuss the numerical simulation for a class of constrained optimal control problems governed by integral equations. The Galerkin method is used for the approximation of the problem. A priori error estimates and a superconvergence analysis for the approximation scheme are presented. Based on the results of the superconvergence analysis, a recovery type a posteriori error estimator is provided, which can be used for adaptive mesh refinement.

Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations

Xu Yin, Waixiang Cao, Zhimin Zhang (2024)

Applications of Mathematics

We present a unified approach to studying the superconvergence property of the spectral volume (SV) method for high-order time-dependent partial differential equations using the local discontinuous Galerkin formulation. We choose the diffusion and third-order wave equations as our models to illustrate approach and the main idea. The SV scheme is designed with control volumes constructed using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes...

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