Extrapolation methods for vector sequences
SIAM Review
The node-centred finite volume approach: bridge between finite differences and finite elements
Computer Methods in Applied Mechanics and Engineering
Computational algorithms for aerodynamic analysis and design
Applied Numerical Mathematics
High-resolution conservative algorithms for advection in incompressible flow
SIAM Journal on Numerical Analysis
A comparative study on methods for convergence acceleration of iterative vector sequences
Journal of Computational Physics
A fast, matrix-free implicity method for compressible flows on unstructured grids
Journal of Computational Physics
Iterative Procedures for Nonlinear Integral Equations
Journal of the ACM (JACM)
Flux correction tools for finite elements
Journal of Computational Physics
Stopping criteria for iterations in finite element methods
Numerische Mathematik
On the design of general-purpose flux limiters for finite element schemes. I. Scalar convection
Journal of Computational Physics
Non-oscillatory third order fluctuation splitting schemes for steady scalar conservation laws
Journal of Computational Physics
Journal of Computational Physics
Explicit and implicit FEM-FCT algorithms with flux linearization
Journal of Computational Physics
Journal of Computational Physics
Failsafe flux limiting and constrained data projections for equations of gas dynamics
Journal of Computational Physics
Anderson Acceleration for Fixed-Point Iterations
SIAM Journal on Numerical Analysis
A flux-corrected transport algorithm for handling the close-packing limit in dense suspensions
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
This paper is concerned with the development of general-purpose algebraic flux correction schemes for continuous (linear and multilinear) finite elements. In order to enforce the discrete maximum principle (DMP), we modify the standard Galerkin discretization of a scalar transport equation by adding diffusive and antidiffusive fluxes. The result is a nonlinear algebraic system satisfying the DMP constraint. An estimate based on variational gradient recovery leads to a linearity-preserving limiter for the difference between the function values at two neighboring nodes. A fully multidimensional version of this scheme is obtained by taking the sum of local bounds and constraining the total flux. This new approach to algebraic flux correction provides a unified treatment of stationary and time-dependent problems. Moreover, the same algorithm is used to limit convective fluxes, anisotropic diffusion operators, and the antidiffusive part of the consistent mass matrix. The nonlinear algebraic system associated with the constrained Galerkin scheme is solved using fixed-point defect correction or a nonlinear SSOR method. A dramatic improvement of nonlinear convergence rates is achieved with the technique known as Anderson acceleration (or Anderson mixing). It blends a number of last iterates in a GMRES fashion, which results in a Broyden-like quasi-Newton update. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convection-dominated transport problems and anisotropic diffusion equations in 2D.