The pointwise stabilities of piecewise linear finite element method on non-obtuse tetrahedral meshes of nonconvex polyhedra

03/09/2021

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by   Huadong Gao, et al.

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Let Ω be a Lipschitz polyhedral (can be nonconvex) domain in ℝ^3, and V_h denotes the finite element space of continuous piecewise linear polynomials. On non-obtuse quasi-uniform tetrahedral meshes, we prove that the finite element projection R_hu of u ∈ H^1(Ω) ∩ C(Ω) (with R_h u interpolating u at the boundary nodes) satisfies ‖ R_h u‖_L^∞(Ω)≤ C |log h |‖ u‖_L^∞(Ω). If we further assume u ∈ W^1,∞(Ω), then ‖ R_h u‖_W^1, ∞(Ω)≤ C |log h |‖ u‖_W^1, ∞(Ω).