The Reference Solution Approach to Hp-Adaptivity in Finite Element Flux-Corrected Transport Algorithms

Abstract

This paper presents an $hp$-adaptive flux-corrected transport algorithm based on the reference solution approach. It features a finite element approximation with unconstrained high-order elements in smooth regions and constrained $Q1$ elements in the neighborhood of steep fronts. The difference between the reference solution and its projection into the current (coarse) space is used as an error indicator to determine the local mesh size $h$ and polynomial degree $p$. The reference space is created by increasing the polynomial degree $p$ in smooth elements and $h$-refining the mesh in nonsmooth elements. The smoothness is determined by a hierarchical regularity estimator based on discontinuous higher-order reconstructions of the solution and its derivatives. The discrete maximum principle for linear/bilinear finite elements is enforced using a linearized flux-corrected transport (FCT) scheme. $p$-refinement is performed by enriching a continuous bilinear approximation with continuous or discontinuous basis functions of polynomial degree $p\ge 2$. The algorithm is implemented in the open-source software package HERMES. The use of hierarchical data structures that support arbitrary level hanging nodes makes the extension of FCT to $hp$-FEM relatively straightforward. The accuracy of the proposed methodology is illustrated by a numerical example for a two-dimensional benchmark problem with a known exact solution.