Gradient recovery techniques for the design of a posteriori error indicators are reviewed in the context
of fluid dynamic problems featuring shocks and discontinuities. An edgewise slope limiting approach
tailored to linear finite element discretizations is presented [1]. The improved gradient values at edge
midpoints are recovered as the limited average of adjacent slopes. Furthermore, the constant gradient
values may serve as natural upper and lower bounds to be imposed on the edge slopes. To this end,
renowned techniques such as averaging projection, the Zienkiewicz-Zhu patch recovery (SPR) and
polynomial preserving recovery (PPR) are used to predict smoothed gradient values which are corrected
subject to geometric constraints by applying a slope limiter edge-by-edge. In either case, a second order
accurate quadrature rule is employed to measure the difference between consistent and reconstructed
gradient values in the (local) L2-norm which provides a usable indicator for grid adaptivity.
The algebraic flux correction (AFC) methodology, presented in [2], is equipped with adaptive mesh
refinement/coarsening procedures governed by the recovery based error indicator. Algebraic criteria are
derived to steer grid improvement techniques such as mesh smoothing and edge swapping approaches.
Finally, the general-purpose flux limiter, recently developed in [3], which lends itself to the treatment
of stationary and time-dependent problems alike is extended to systems of nonlinear conservation laws.
The adaptive algorithm is applied to inviscid compressible flows at different Mach numbers.
REFERENCES
[1] M. M¨oller and D. Kuzmin. Adaptive mesh refinement for high-resolution finite element
schemes. Technical report 297, University of Dortmund, 2005. To appear in: Int. J. Numer.
Meth. Fluids.
[2] D. Kuzmin and M. M¨oller, Algebraic flux correction II. Compressible Euler Equations. In:
D. Kuzmin, R. L¨ohner and S. Turek (eds.) Flux-Corrected Transport: Principles, Algorithms,
and Applications. Springer, 207-250, 2005.
[3] D. Kuzmin. On the design of general-purpose flux limiters for implicit FEM with a consistent
mass matrix. Technical report 283, University of Dortmund, 2005. Submitted to: J.
Comput. Phys..