We present families of scalar nonconforming finite elements of arbitraryorder $r\ge 1$ with optimal approximation properties on quadrilaterals andhexahedra. Their vector-valued versions together with a discontinuouspressure approximation of order $r-1$ form inf-sup stable finite element pairsof order r for the Stokes problem. The well-known elements by Rannacherand Turek are recovered in the case r=1. A numerical comparison betweenconforming and nonconforming discretisations will be given. Since higherorder nonconforming discretisation on quadrilaterals and hexahedra have lessunknowns and much less non-zero matrix entries compared to correspondingconforming methods, these methods are attractive for numerical simulations.

In this work we consider the dual-primal Discontinuous Petrov–Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous,a static condensation procedure can becarried out, leading to a single-field nonconformingdiscretization scheme. For this latter formulation,we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. A convergence analysis isdeveloped, proving first-order accuracy of the method in a discrete H 1-norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers.

For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution.Thus, for the error analysis, the existing theory from the conformingcase can be used together with some simple additional arguments.As an essential point, the property is exploited that the nonconformingfinite element space contains as a subspace a conforming finite element space of first order. This property is fulfilled for many knownnonconforming spaces. We prove local lower and global upper a posteriori error estimates for an enhanced error measure which is the discretization error in the discrete energy norm plus the error of the best representation of the exact solution by a function in the conforming space used for thepost-processing. We demonstrate that the idea to use a computed conforming approximation ofthe nonconforming solution can be applied also to derive an a posteriorierror estimate for a linear functional of the solution which representssome quantity of physical interest.