Estimates for Piecewise Linear Approximation of Derivative Functions of Sobolev Classes

Abstract

The article considers the problem of estimating the error in calculating the gradient of functions of Sobolev classes for piecewise linear approximation on triangulations. Traditionally, problems of this kind are considered for continuously differentiable functions. In this case, the corresponding estimates reflect both the smoothness class of the functions and the quality of the triangulation simplexes. However, in problems of substantiating the existence of a solution to a variational problem in the nonlinear theory of elasticity, conditions arise for permissible deformations in terms of generalized derivatives. Therefore, for the numerical solution of these problems, conditions are required that provide the necessary approximation of the derivatives of continuous functions of the Sobolev classes. In this article, an integral estimate of the indicated error for continuously differentiable functions is obtained in terms of the norms of the corresponding spaces for functions that reflect, on the one hand, the quality of triangulation of the polygonal region, and on the other hand, the class of functions with generalized derivatives. The latter is expressed in terms of the majorant of the modulus of continuity of the gradient. To obtain the final estimate, the possibility of passing to the limit using the norm of the Sobolev space is proven.

Keywords

• triangulation
• delaunay triangulation
• piecewise linear approximation
• gradient approximation
• numerical methods