Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods
SIAM Journal on Numerical Analysis
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
On the stability of the L2 projection in H1(Ω)
Mathematics of Computation
Compactness Properties of the DG and CG Time Stepping Schemes for Parabolic Equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
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Suppose S ⊂ H1(Ω) is a finite-dimensional linear space based on a triangulation T of a domain Ω, and let Π : L2(Ω) → L2 (Ω) denote the L2-projection onto S. Provided the mass matrix of each element T ∈ T and the surrounding mesh-sizes obey the inequalities due to Bramble, Pasciak, and Steinbach or that neighboring element-sizes obey the global growth-condition due to Crouzeix and Thomée, Π is H1-stable: For all u ∈ H1 (Ω) we have ||Πu||H1 (Ω) ≤ C ||u||H1 (Ω) with a constant C that is independent of, e.g., the dimension of S.This paper provides a more flexible version of the Bramble-Pasciak-Steinbach criterion for H1-stability on an abstract level. In its general version, (i) the criterion is applicable to all kind of finite element spaces and yields, in particular, H1-stability for nonconforming schemes on arbitrary (shape-regular) meshes; (ii) it is weaker than (i.e., implied by) either the Bramble-Pasciak-Steinbach or the Crouzeix-Thomée criterion for regular triangulations into triangles; (iii) it guarantees H1-stability of Π a priori for a class of adaptively-refined triangulations into right isosceles triangles.