An hp-adaptive flux-corrected transport algorithm for continuous finite elements

摘要

This paper presents an (hp)-adaptive flux-corrected transport algorithm for continuous finite elements. The proposed approach is based on a continuous Galerkin approximation with unconstrained higher-order elements in smooth regions and constrained (P_1/Q_1) elements in the neighborhood of steep fronts. Smooth elements are found using a hierarchical smoothness indicator based on discontinuous higher-order reconstructions. A gradient-based error indicator determines the local mesh size (h) and polynomial degree (p). The discrete maximum principle for linear/bilinear finite elements is enforced using a linearized flux-corrected transport (FCT) algorithm. The same limiting strategy is employed when it comes to constraining the (L^2) projection of data from one finite-dimensional space into another. The new 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 method is illustrated by a numerical study for a two-dimensional benchmark problem with a known exact solution.