H^1-Stability of the L^2-Projection onto Finite Element Spaces on Adaptively Refined Quadrilateral Meshes

08/28/2020
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by   Mazen Ali, et al.
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The L^2-orthogonal projection Ī _h:L^2(Ī©)ā†’š•_h onto a finite element (FE) space š•_h is called H^1-stable iff āˆ‡Ī _h u_L^2(Ī©)ā‰¤ Cu_H^1(Ī©), for any uāˆˆ H^1(Ī©) with a positive constant Cā‰  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,ā€¦,9.

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