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 Ω ⊂ ℜ 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 ∂ Ξ 1 ( u ) / ∂ z + ⋯ + g 4 ∂ Ξ 4 ( u ) / ∂ z of the constant derivatives of the linear interpolants Ξ 1 ( u ) , … , Ξ 4 ( u ) of u in the vertices of the triangles U 1 = a c 1 c 2 ¯ , … , U 4 = a c 4 c 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 ( Ω ¯ ) , 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.