Maximum principle and uniform convergence for the finite element method

The solution of the Dirichlet boundary value problem over a polyhedral domain Ω ⊂ Rⁿ, n ≥ 2, associated with a second-order elliptic operator, is approximated by the simplest finite element method, where the trial functions are piecewise linear. When the discrete problem satisfies a maximum principle, it is shown that the approximate solution u_h converges uniformly to the exact solution u if u ε{lunate} W_1,p (Ω), with p > n, and that ∥u-u_h∥_L∞(Ω) = O(h) if u ε{lunate} W²,p(Ω), with 2p > n. In the case of the model problem -Δu+au = f in Ω, u = u_o on δΩ, with a ≥ 0, a simple geometrical condition is given which insures the validity of the maximum principle for the discrete problem. © 1973.