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A dual finite element complex on the barycentric refinement. (English) Zbl 1130.65108
Given a $2$-dimensional oriented surface equipped with a simplicial mesh, the standard finite elements of lowest order provide a complex $$P_1\text{ elements }@>\text{curl}>>\text{ Raviart-Thomas space }RT_0@>\text{div}>>P_0\text{ elements.}$$ A new complex of finite element spaces on the barycentric refinement of the mesh is constructed that yield a discrete analogue of $$H^1@>\text{grad}>> H(\text{curl})@>\text{curl}>>L^2.$$ The only difference is that the functions in the middle space are rotated by the operation $u\to u\times n$. The resulting spaces are $L^2$-dual to the given ones and satisfy an inf-sup condition with constants that are bounded from below for all $h$.

65N12Stability and convergence of numerical methods (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N50Mesh generation and refinement (BVP of PDE)
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