On the performance of high-order finite elements with respect to maximum principles and the nonnegative constraint for diffusion-type equations

SUMMARY The main aim of this paper is to document the performance of p‐refinement with respect to maximum principles and the nonnegative constraint. The model problem is steady‐state anisotropic diffusion with decay (which is a second‐order elliptic partial differential equation). We consider the standard single‐field formulation based on the Galerkin formalism and two least squares‐based mixed formulations. We employ Lagrange polynomials with unequal‐spaced points, and polynomials of order p = 1 to p = 10 are used. It is shown that the violation of the nonnegative constraint cannot be overcome with p‐refinement alone for anisotropic diffusion. We shall illustrate the performance of p‐refinement by using several representative problems. The intended outcomes of the paper are twofold. First, this study will caution the users of high‐order approximations about their performance with respect to maximum principles and the nonnegative constraint. Second, this study will help researchers develop new methodologies for enforcing maximum principles and the nonnegative constraint under high‐order approximations. Copyright © 2012 John Wiley & Sons, Ltd.