Finite Element Complexes in Two Dimensions
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Two-dimensional finite element complexes with various smoothness, including the de Rham complex, the curldiv complex, the elasticity complex, and the divdiv complex, are systematically constructed in this work. First smooth scalar finite elements in two dimensions are developed based on a non-overlapping decomposition of the simplicial lattice and the Bernstein basis of the polynomial space. Smoothness at vertices is more than doubled than that at edges. 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 elasticity complexes and finite element divdiv complexes are derived from finite element de Rham complexes by using the Bernstein-Gelfand-Gelfand (BGG) framework. Additionally, some finite element divdiv complexes are constructed without BGG framework. Dimension count plays an important role for verifying the exactness of two-dimensional finite element complexes.
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