We construct 2D and 3D finite element de Rham sequences of arbitrary polynomial degrees with extra smoothness. Some of these elements have nodal degrees of freedom and can be considered as generalisations of scalar Hermite and Lagrange elements. Using the nodal values, the number of global degrees of freedom is reduced compared with the classical Nédélec and Brezzi–Douglas–Marini finite elements, and the basis functions are more canonical and easier to construct. Our finite elements for H(div) with regularity r = 2 coincide with the nonstandard elements given by Stenberg (Numer Math 115(1):131–139, 2010).We show how regularity decreases in the finite element complexes, so that they branch into known complexes. The standard de Rham complexes of Whitney forms and their higher order version can be regarded as the family with the lowest regularity.  We also talk about the  treatment of inconsistent boundary conditions for this kind of elements. 

胡俊本科毕业于湘潭大学, 曾于2004年获得德国洪堡基金会的洪堡研究奖学金, 于2006年获得教育部全国百篇优秀博士学位论文。博士毕业后加入北京大学数学科学学院并任教至今。胡俊的研究得到了国家自然科学基金委重大研究计划重点项目、创新群体、重点项目与面上项目等多个项目经费资助。胡俊教授一直致力于有限元方法的研究, 由于他的杰出工作, 北京国际数学研究中心2014年特邀他为访问教授; 2015年被中国计算数学学会授予首届“青年创新奖”, 以表彰他对弹性力学方程组混合有限元方法所作出的奠基性贡献; 2016年国家杰出青年基金获得者。