Hölder estimates for Green's functions on convex polyhedral domains and their applications to finite element methods
A model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The authors first obtain sharp Hölder estimates for the corresponding Green's function. Applying these to finite element methods, the best approximation property of the error in \(W_\infty^1\) is shown [see \textit{S. C. Brenner} and \textit{R. L. Scott}, The mathematical theory of finite element methods. 3rd ed. Texts in Applied Mathematics 15. (New York), NY: Springer. (2008; Zbl 1135.65042)]. It is also observed that using local estimates obtained by \textit{Z. Chen} and \textit{Z. Chen} [SIAM J. Numer. Anal. 42, 1146--1166 (2004; Zbl 1081.65102)] and \textit{J. Guzman} [Math. Comp. 75, No.~255, 1067--1085 (2006; Zbl 1092.65096)] and the techniques used here, one can prove optimal \(W_\infty^1\) error estimates for various discontinuous Galerkin methods on convex polyhedral domains.
scientific article; zbMATH DE number 5547474
scientific article