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

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

Assistant Teacher, Department of Computer Science and Experimental Mathematics
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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 Tm.

Suppose where dS is a length of the maximum side of a triangle STm.

Let f : DD*, D*R2 be a mapping such that f(x) = (U(x), V(x)), xD, x= (x1, x2), where U, VC2(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 Tm we construct an approximating mapping fm: DD*, fm(x) = (Um(x), Vm(x)), such that Um, Vm are piecewise smooth functions and

Suppose dxf is the differential of f at the point xD, Jf (x) is the Jacobian matrix of f at the point x.
For every STm we construct a mapping Am(x), xS that approximates dxf using functions Um, Vm 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 fm: D D*, fm(x) = (Um(x), Vm(x)), such that Um, Vm are piecewise smooth functions and

for any pPmδ > 0.

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