Papers › H¹-Stability of the L²-Projection onto Finite Element Spaces on Adaptively Refined...
H¹-Stability of the L²-Projection onto Finite Element Spaces on Adaptively Refined Quadrilateral Meshes
Mazen Ali, Stefan A. Funken, Anja Schmidt
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The L²-orthogonal projection Πₕ:L²(Ω)→𝕍ₕ onto a finite element (FE) space 𝕍ₕ is called H¹-stable iff ∇Πₕ u_(L²(Ω))≤Cu_(H¹(Ω)), for any u∈H¹(Ω) with a positive constant C≠C(h) independent of the mesh size h>0. In this work, we discuss local criteria for the H¹-stability of adaptively refined meshes. We show that adaptive refinement strategies for quadrilateral meshes in 2D (Q-RG and Q-RB), introduced originally in Bank et al. 1982 and Kobbelt 1996, are H¹-stable for FE spaces of polynomial degree p=2,…,9.
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