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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 \(h\).

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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