Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

作者：Dalik Josef^*; Valenta Vaclav

来源：Central European Journal of Mathematics, 2013, 11(4): 597-608.

DOI：10.2478/s11533-012-0159-7

• Linear triangular and bilinear rectangular finite element
• Nonobtuse regular triangulation
• Averaging partial derivatives
• A posteriori error estimator
• Adaptive solution of elliptic differential problems in 2D

An averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x (1), x (2)) at the vertices of a regular triangulation T (h) composed both of rectangles and triangles is presented. The method assumes that only the interpolant I (h) [u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T (h) is known. A complete analysis of this method is an extension of the complete analysis concerning the finite element spaces of linear triangular elements from [Dalik J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619-644]. The second-order approximation of the gradient is extended from the vertices to the whole domain and applied to the a posteriori error estimates of the finite element solutions of the planar elliptic boundary-value problems of second order. Numerical illustrations of the accuracy of the averaging method and of the quality of the a posteriori error estimates are also presented.

• 出版日期2013-4