Reduced averaging of directional derivatives in the vertices of unstructured triangulations

Assume that Th is a conforming regular triangulation without obtuse angles of a bounded polygonal domain Ω⊂ℜ2. 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 Th only, is presented. In the general case, the construction consists of (a) the choice of a special five-tuple c1,…,c5 of neighbours of a and (b) the solution of a system of four equations in the unknowns g1,…,g4 guaranteeing that the linear combination R[z,u](a)=g1∂Ξ1(u)/∂z+⋯+g4∂Ξ4(u)/∂z of the constant derivatives of the linear interpolants Ξ1(u),…,Ξ4(u) of u in the vertices of the triangles 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(h2) for all , 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.