Numerical Analysis and Its Applications: 4th International by Mikhail V. Borsuk (auth.), Svetozar Margenov, Lubin G.

By Mikhail V. Borsuk (auth.), Svetozar Margenov, Lubin G. Vulkov, Jerzy Waśniewski (eds.)

This publication constitutes the completely refereed post-conference court cases of the 4th foreign convention on Numerical research and Its functions, NAA 2008, held in Lozenetz, Bulgaria in June 2008.

The sixty one revised complete papers provided including thirteen invited papers have been rigorously chosen in the course of rounds of reviewing and development. The papers handle all present elements of numerical research and talk about a variety of difficulties referring to fresh achievements in physics, chemistry, engineering, and economics. a different concentration is given to numerical approximation and computational geometry, numerical linear algebra and numerical resolution of transcendental equations, numerical tools for differential equations, numerical modeling, and excessive functionality medical computing.

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Additional resources for Numerical Analysis and Its Applications: 4th International Conference, NAA 2008, Lozenetz, Bulgaria, June 16-20, 2008. Revised Selected Papers

Example text

Then for every n ∈ N, the zeros of pn are simple. Proof. This proof is due to the fact that the weight wJ (x) = (1 − x)α (1 + x)β , is classical. The weight function w is classical provided there exist two polynomials φ and ψ, of degrees at most two and one, respectively, such that (φwJ )′ (x) = Numerical Integration with Complex Jacobi Weight Function 23 (ψwJ )(x), for x in the support of the measure µJ . In this case we can check directly that φ(x) = 1 − x2 , ψ(x) = −(1 + α)(1 + x) + (1 + β)(1 − x).

We denote by Yh the affine set of functions in Xh interpolating the data Yh = {u ∈ Xh : u(x) = dh (x), ∀x ∈ Vh } . Our goal is now to find a function in Yh that oscillates as little as possible. We think of such a function as one that best fits the data map dh . For this purpose, we introduce the following functional (|ux x | + 2|ux y | + |uy y |) + α Jh (u) = T ∈Th T |[[un ]]| , u ∈ Xh F F ∈Fhi representing the total variation of the gradient of u with a weight, α > 0. Note that Jh defines a semi-norm which vanishes if and only if its argument is a linear function on Ω.

The novelty of the approach in the present paper is to relax the C 1 -smoothness on the finite element space which is used in the data reconstruction process. The discrete space is S. G. Vulkov, and J. ): NAA 2008, LNCS 5434, pp. 32–43, 2009. c Springer-Verlag Berlin Heidelberg 2009 Surface Reconstruction via L1 -Minimization 33 composed of continuous finite elements with possibly discontinuous gradients. This is the natural discretization setting for functions that are in W 1,1 and whose gradient has bounded total variation.

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