The algebraic flux correction methodology, as introduced in [1], is embedded
into an adaptive mesh refinement algorithm for scalar conservation laws. Flux
limiters have been successfully used to construct high-resolution finite element
schemes for convection dominated problems. Starting with a linear discretization
of high order, such methods gradually turn into a nonoscillatory low-order
scheme in the vicinity of steep gradients and discontinuities of the solution
profile. Primarily designed to modulate the antidiffusive correction to the
overly diffusive low-order scheme, flux limiters implicitly detect zones of the
underlying mesh where the flow structure is not accurately resolved and, thus,
must be `modeled` by numerical diffusion. Hence, the /emph{nodal} ratio of
nonlinear artificial (anti-)diffusion and high-order convective contributions,
which is computable at no additional cost, provides a usable error indicator.

The extension of the algebraic flux correction methodology to systems of
nonlinear conservation laws, as described in [2], is combined with the new
limiter-based grid refinement strategy and applied to the compressible Euler
equations. The nonlinear algebraic system is solved by a preconditioned defect
correction scheme. As a generalization of Roe`s approximate Riemann solver, the
underlying low-order scheme is derived by rendering all off-diagonal matrix
blocks positive semi-definite. The preconditioner is assembled edge-by-edge
making use of scalar artificial viscosity proportional to the spectral radius of
the Roe.  The limiting process is performed in terms of local characteristic
variables. The node-oriented limiter is applied to the vector of fluxes in each
coordinate direction which allows for anisotropic mesh refinement. Various
algorithmic aspects including the implementation of characteristic boundary
conditions are addressed. Numerical examples are presented for both scalar
convection-diffusion problems and the compressible Euler equations at different
Mach numbers.