Finite Elements for div- and divdiv-Conforming Symmetric Tensors in Arbitrary Dimension

Abstract

Several div-conforming and divdiv-conforming finite elements for symmetric tensors on simplexes in arbitrary dimension are constructed in this work. The shape function space is first split as the trace space and the bubble space. The later is further decomposed into the null space of the differential operator and its orthogonal complement. Instead of characterizations of these subspaces of the shape function space, characterizations of corresponding degrees of freedom in the dual spaces are provided. Vector div-conforming finite elements are first constructed as an introductory example. Then new symmetric div-conforming finite elements are constructed. The dual subspaces are then used as build blocks to construct new divdiv-conforming finite elements.