Finite element complexes with various smoothness, including the de Rham complex, the Stokes complex, the curldiv complex, the Hessian complex, the elasticity complex, and the divdiv complex, are systematically constructed in this work. First smooth scalar finite elements are developed based on a non-overlapping decomposition of the simplicial lattice and the Bernstein basis of the polynomial space. Smoothness at vertices and on edges are more than doubled than those on edges and on faces, respectively. Then the finite element de Rham complexes with various smoothness are devised using smooth finite elements with smoothness parameters satisfying certain relations. Finally, finite element Hessian complexes, finite element elasticity complexes and finite element divdiv complexes are derived from finite element de Rham complexes by using the Bernstein-Gelfand-Gelfand (BGG) framework. Dimension count and div stability play an important role for verifying the exactness of finite element complexes. 

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。学术论文方面，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。科研课题方面，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)。