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An $$hp$$-adaptive Flux-corrected Transport Algorithm For Continuous Finite Elements

Melanie Bittl, D. Kuzmin
Published 2012 · Mathematics

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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.
This paper references
Fully multidimensional flux-corrected transport algorithms for fluids
S. Zalesak (1979)
A simple error estimator and adaptive procedure for practical engineerng analysis
O. C. Zienkiewicz (1987)
Scalar Conservation Laws
R. LeVeque (1992)
High-resolution conservative algorithms for advection in incompressible flow
R. LeVeque (1996)
Flux correction tools for finite elements
D. Kuzmin (2002)
Higher-Order Finite Element Methods
I. Doležel (2003)
Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method
T. Davis (2004)
Higher-order finite element methods
R. Lazarov (2005)
Flux-Corrected Transport
D. Kuzmin (2005)
Algebraic Flux Correction I. Scalar Conservation Laws
D. Kuzmin (2005)
Automatic hp-Adaptivity With Arbitrary-Level Hanging Nodes
Sol ´ õn (2006)
Automatic hp-Adaptivity With Arbitrary-Level Hanging Nodes
P. Solín (2006)
libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations
B. Kirk (2006)
Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM
P. Solín (2008)
On the design of algebraic flux correction schemes for quadratic finite elements
D. Kuzmin (2008)
Finite element methods for time-dependent convection – diffusion – reaction equations with small diffusion
V. John (2008)
Failsafe flux limiting and constrained data projections for equations of gas dynamics
D. Kuzmin (2010)
A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods
D. Kuzmin (2010)
Adaptive hierarchic transformations for dynamically p-enriched slope-limiting over discontinuous Galerkin systems of generalized equations
C. Michoski (2011)
Automatic hp-adaptivity on Meshes with Arbitrary-Level Hanging Nodes in 3D
P. Kus (2011)
Algebraic Flux Correction II
D. Kuzmin (2012)
Hermes-higher-order modular finite element system (user's guide)
P Solin (2012)
A parameter-free smoothness indicator for high-resolution finite element schemes
D. Kuzmin (2013)
Slope limiting for discontinuous Galerkin approximations with a possibly non‐orthogonal Taylor basis
D. Kuzmin (2013)
M Bittl

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