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.
黄学海，上海财经大学讲席教授、博士研究生导师，研究方向为有限元方法。在Math. Comp.、SIAM J. Numer. Anal.、Numer. Math.、J. Sci. Comput.等国际期刊发表SCI论文三十多篇。正主持一项国家自然科学基金面上项目和上海市自然科学基金原创探索项目，主持完成国家自然科学基金面上项目、青年项目、数学天元项目和温州市科技计划项目各一项、浙江省自然科学基金项目两项。获中国计算数学学会优秀青年论文竞赛优秀奖，博士学位论文被评为上海市研究生优秀成果(学位论文)。