Boluchevskaya A.V. Approximation of differentials of elliptic systems solutions

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http://dx.doi.org/10.15688/jvolsu1.2013.1.3

Boluchеvskaya Anna Vladimirovna

Assistant Teacher, Department of Computer Science and Experimental Mathematics
Volgograd State University
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Prospekt Universitetskij, 100, 400062 Volgograd, Russian Federation

Abstract. We consider the elliptic system of differential equations and construct piecewise smooth approximation of its solution using precise and approximate values at the triangulation nodes. These piecewise smooth mappings are used to approximate the differential of the solution with an error that does not depend on the level of triangles degeneracy.

    Let be an array of finite sets of points in the domain D ⊂ R². For every set we consider its triangulation T_m.

     Suppose where dS is a length of the maximum side of a triangle S ∈ T_m.

     Let f : D → D^*, D^* ⊂ R² be a mapping such that f(x) = (U(x), V(x)), x ∈ D, x= (x₁, x₂), where U, V ∈ C²(D) are the solutions of the elliptic system

a = a(x), b = b(x), c = c(x), d = d(x) ∈ C¹(D),  in D.

    For every T_m we construct an approximating mapping f_m: D → D^*, f_m(x) = (U_m(x), V_m(x)), such that U_m, V_m are piecewise smooth functions and

     Suppose d_xf is the differential of f at the point x ∈ D, J_f (x) is the Jacobian matrix of f at the point x.
    For every S ∈ T_m we construct a mapping A_m(x), x ∈ S that approximates d_xf using functions U_m, V_m and coefficients of the elliptic system. We also show that the approximation error

does not depend on the level of the triangle S degeneracy.
     The same results are obtained if we use an approximating mapping f_m: D → D^*, f_m(x) = (U_m(x), V_m(x)), such that U_m, V_m are piecewise smooth functions and

for any p ∈ P_m, δ > 0.

Key words: piecewise smooth approximation, approximation of the differential, triangulation, elliptic system of equations, approximation error.

Approximation of differentials of elliptic systems solutions by Boluchevskaya A.V. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №1 (18) 2013 pp. 31-44

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