H^1-Stability of the L^2-Projection onto Finite Element Spaces on Adaptively Refined Quadrilateral Meshes
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The L^2-orthogonal projection Π_h:L^2(Ω)→𝕍_h onto a finite element (FE) space 𝕍_h is called H^1-stable iff ∇Π_h u_L^2(Ω)≤ Cu_H^1(Ω), for any u∈ H^1(Ω) with a positive constant C≠ C(h) independent of the mesh size h>0. In this work, we discuss local criteria for the H^1-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 H^1-stable for FE spaces of polynomial degree p=2,…,9.
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