The algebraic flux-correction (AFC) approach introduced in [8, 17] for the accurate
treatment of convection-dominated flow problems and refined in a series of publications
[9,11–15,18,19] is extended to non-conforming finite element discretizations. Originally,
this class of multidimensional high-resolution schemes was developed in the framework of conforming
(multi-)linear P1=Q1 approximations. The underlying design criteria are revisited for
non-conforming approximations on unstructured quadrilateral meshes. The properties of (non-
)parametric rotated multi-linear Qrot
1 finite elements [23] are analyzed. The midpoint-value
based variant Qrot;MP
1 is shown to violate essential design criteria unconditionally. The alternative
definition of local basis functions in terms of integral mean values is shown to comply with
the concepts of algebraic flux correction. A high-resolution continuous Galerkin scheme for
Qrot;MV
1 finite elements is presented and investigated numerically for a two-dimensional benchmark
problem with known exact solution. The advantages of the proposed non-conforming
approach over P1=Q1 approximations are discussed. The regular structure of system matrices
is the basis for using efficient sparse matrix formats such as ELLPACK on fully unstructured
meshes. Further implementation details including aspects of parallelization are adressed.