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Nested tetrahedral grids and strengthened C. B. S. inequality. (English) Zbl 1071.65164

The paper deals with nested two-level decompositions of 3D domains into tetrahedra and with the corresponding spaces of continuous piecewise linear finite element (FE) functions. In detail, a FE space \(V_ h\) corresponds to a fine grid characterized by a discretization parameter \(h\), and a FE space \(V_ H\) corresponds to a coarse grid characterized by a parameter \(H\). Also, a FE space \(V_ h^ 0\), a complement to \(V_ H\) in \(V_ h\), is introduced. For a symmetric positive bilinear form \(a\), a constant \(\gamma \) appears in the strengthened Cauchy-Bunyakowski-Schwarz inequality \[ | a(u,v)| \leq \gamma \sqrt {a(u,u)}\sqrt {a(v,v)} \quad \forall u\in V_ H\;\;\forall v\in V_ h^ 0. \] The constant \(\gamma \) is estimated for \(a\) corresponding to various operators; namely an anisotropic Laplacian and an anisotropic elasticity operator. The estimates are expressed in terms of \(m=H/h\). The knowledge of \(\gamma \) is important in the convergence analysis of multigrid methods, multi-level preconditioners, or composite grid iterative methods. For triangular elements in 2D, estimates of \(\gamma \) can be found in O. Axelsson and R. Blaheta [Appl. Math. 49, 57–72 (2004)].

MSC:

65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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References:

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