Reduced averaging of directional derivatives in the vertices of unstructured triangulations

Assume that T"h is a conforming regular triangulation without obtuse angles of a bounded polygonal domain @W@?@?^2. For an arbitrary unit vector z and an inner or so-called semi-inner vertex a, the method of reduced averaging for the approximation of the derivative @?u/@?z(a) of a smooth function u, known in the vertices of T"h only, is presented. In the general case, the construction consists of (a) the choice of a special five-tuple c^1,...,c^5 of neighbours of a and (b) the solution of a system of four equations in the unknowns g"1,...,g"4 guaranteeing that the linear combination R[z,u](a)=g"1@?@X"1(u)/@?z+...+g"4@?@X"4(u)/@?z of the constant derivatives of the linear interpolants @X"1(u),...,@X"4(u) of u in the vertices of the triangles U"1=ac^1c^2@?,...,U"4=ac^4c^5@? satisfies R[z,u](a)=@?u/@?z(a) for all quadratic polynomials u. The approximations R[z,u](a) are proved to be of the accuracy O(h^2) for all u@?C^3(@W@?), shown to be more effective than the local approximations of @?u/@?z(a) by the other known second-order operators and compared with them numerically.