Klyachin A.A. On piecewise-linear almost-solutions of elliptic equations

Klyachin Alеksеy Alеksandrovich

Doctor of Physical and Mathematical Sciences, Head of Department of Mathematical Analysis and Function Theory Volgograd State University
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Prospect Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. In this paper we define deviation εQ (N) of piecewise-linear almost-solution N of the minimal surface equation Q[f(x)] = 0 and we get a general formula to calculate it.  Let  be tetrahedrons which have vertex Pi. We denote sides leaving from vertex Pi of the tetrahedron , = 1, 2, ..., k(i), and let be  the side of tetrahedron opposite to vertex Pi. We set by the external normal vectors of the sides relatively of  . As N is linear function in  then ∇N ≡ const. Then the following equality holds

 

where  is exterior side relatively Pi. On the basis of this concept it obtained approximation equation εQ(fN) = 0 or μijuij=(μij,5 + μij,6)ui+1j + (μij,2μij,3)ui-1j + (μij,1 + μij,2)uij+1 + (μij,4 + μij,5)uij-1,

and proved that the deviation εQ(uN) converges to the integral of the modulus of the mean curvature of the graph of C2-smooth function u, that is

     Thus, the obtained system of nonlinear equations aproximate the minimal surface
equation.

Key words: piecewise-linear functions, almost-solution, minimal surface equation, approximation equation, deviation of piecewise-linear almost-solution.

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On piecewise-linear almost-solutions of elliptic equations by Klyachin A.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №2 (19) 2013 pp. 18-25

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