Weak discrete maximum principle of finite element methods in convex polyhedra

09/15/2019

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by   Dmitriy Leykekhman, et al.

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We prove that the Galerkin finite element solution u_h of the Laplace equation in a convex polyhedron , with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree r> 1, satisfies the following weak maximum principle: u_h_L^∞()< Cu_h_L^∞(∂) , with a constant C independent of the mesh size h. By using this result, we show that Ritz projection operator R_h is stable in L^∞ norm uniformly in h for r≥ 2, i.e. R_hu_L^∞()< Cu_L^∞() . Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.