Speaker：Xuehai Huang（Shanghai University of Finance and Economics）

Time：2022-11-04, 10:00

Location：Tencent Meeting ID：243507140（No Password）

Abstract:

Several div-conforming and divdiv-conforming finite elements for symmetric tensors on simplexes in arbitrary dimension are constructed. 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. Furthermore, we consider geometric decompositions of H(div)-conforming finite element tensors. A unified construction of H(div)-conforming finite elements, including vector element, symmetric matrix element, and traceless matrix element, is developed. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Then the vector or matrix at each sub-simplex is decomposed into the tangential and the normal components. The tangential component forms the polynomial bubble function space and the normal component characterizes the trace of div operator. Some degrees of freedom on the normal component can be redistributed facewisely. Discrete inf-sup conditions are verified. This is a joint work with Long Chen from University of California at Irvine.