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The main aim of this paper is to document the performance of $p$-refinementwith respect to maximum principles and the non-negative constraint. The modelproblem is (steady-state) anisotropic diffusion with decay (which is asecond-order elliptic partial differential equation). We considered thestandard single-field formulation (which is based on the Galerkin formalism)and two least-squares-based mixed formulations. We have employed non-uniformLagrange polynomials for altering the polynomial order in each element, and wehave used $p = 1, ..., 10$.  It will be shown that the violation of the non-negative constraint will notvanish with $p$-refinement for anisotropic diffusion. We shall illustrate theperformance of $p$-refinement using several representative problems. Theintended outcome of the paper is twofold. Firstly, this study will caution theusers of high-order approximations about its performance with respect tomaximum principles and the non-negative constraint. Secondly, this study willhelp researchers to develop new methodologies for enforcing maximum principlesand the non-negative constraint under high-order approximations.

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