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Conforming discrete de Rham complexes consisting of finite element spaces with extra smoothness are constructed. In particular, we develop H 2 , H 1 (curl), H 1 and L 2 conforming finite element spaces and show that an exactness property is satisfied. These results naturally lead to discretizations for Stokes and Brinkman type problems as well as conforming approximations to fourth order curl problems. In addition, we reduce the question of stability of the three-dimensional Scott-Vogelius finite element to a simply stated conjecture.doi:10.1090/s0025-5718-2015-02958-5 fatcat:reokeqwpv5b53bkd2bxdk6ml2m