MATHEMATICS OF COMPUTATION Volume 76, Number 260, October 2007, Pages 1833–1846 S 0025-5718(07)02022-4 Article electronically published on April 30, 2007 DISCRETE MAXIMUM PRINCIPLE FOR HIGHER-ORDER FINITE ELEMENTS IN 1D TOM ´ A ˇ S VEJCHODSK ´ Y AND PAVEL ˇ SOL ´ IN Abstract. We formulate a sufficient condition on the mesh under which we prove the discrete maximum principle (DMP) for the one-dimensional Poisson equation with Dirichlet boundary conditions discretized by the hp-FEM. The DMP holds if a relative length of every element K in the mesh is bounded by a value H ∗ rel (p) ∈ [0.9, 1], where p ≥ 1 is the polynomial degree of the element K. The values H ∗ rel (p) are calculated for 1 ≤ p ≤ 100. 1. Introduction Classical (continuous) maximum principles belong to the most important results in the theory of second-order partial differential equations (PDEs). Their discrete counterparts, discrete maximum principles (DMP), appeared in the early 1970s. They were used by various authors to prove the convergence of the lowest-order finite difference and finite element methods (see, e.g., [3, 4] and the references therein). DMP have been studied intensively during the past decades in the context of linear PDEs [2, 8, 10, 17, 18, 20] and more recently also nonlinear equations [9]. Most of these results have two points in common: • they are limited to lowest-order approximations, • they are based on M -matrices [6, 16]. Much less is known about the DMP for methods of higher orders of accuracy such as higher-order finite difference methods, spectral FEM, or hp-FEM. Let us mention, e.g., a result [21] on higher-order collocation methods. Particularly noteworthy is a negative result [7] from 1981 stating that a stronger DMP is not valid for cubic and higher-order Lagrange elements in 2D. In the quadratic case, the stronger DMP is valid under extremely restrictive assumptions on the mesh, which almost never could be satisfied in practice. In light of this negative result, a few attempts were made to formulate and prove weakened forms of the DMP (see, e.g., [11, 14]). The present result is based on the analysis of the discrete Green’s function (DGF) for higher-order elements. A similar concept was used in the piecewise-linear case in [5]. The paper is organized as follows. In Section 2 we introduce the one-dimensional Poisson problem, its hp-FEM discretization, and the discrete maximum principle. Received by the editor January 31, 2006 and, in revised form, July 25, 2006. 2000 Mathematics Subject Classification. Primary 65N30; Secondary 35B50. Key words and phrases. Discrete maximum principle, discrete Green’s function, higher-order elements, hp-FEM, Poisson equation. c 2007 American Mathematical Society Reverts to public domain 28 years from publication 1833 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use