Gradient-Based Limiting and Stabilization of Continuous Galerkin Methods
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In this paper, we stabilize and limit continuous Galerkin discretizations of a linear transport equation using an algebraic approach to derivation of artificial diffusion operators. Building on recent advances in the analysis and design of edge-based algebraic flux correction schemes for singularly perturbed convection-diffusion problems, we derive algebraic stabilization operators that generate nonlinear high-order stabilization in smooth regions and enforce discrete maximum principles everywhere. The correction factors for antidiffusive element or edge contributions are defined in terms of nodal gradients that vanish at local extrema. The proposed limiting strategy is linearity-preserving and provides Lipschitz continuity of constrained terms. Numerical examples are presented for two-dimensional test problems.
KeywordsHyperbolic conservation laws Finite element methods Discrete maximum principles Algebraic flux correction Linearity preservation
This research was supported by the German Research Association (DFG) under grant KU 1530/23-1. The author would like to thank Christoph Lohmann (TU Dortmund University) for helpful discussions and suggestions.
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