by Matthies, G., Schieweck, F..

**Series:** 2007-06, Preprints

- MSC:
- 65N15 Error bounds
- 65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
- 65N50 Mesh generation and refinement

**Abstract:**

For a general diffusion-convection-reaction equation,

we analyze families of nonconforming

finite elements of arbitrary order on a sequence of multilevel grids

consisting of quadrilaterals or hexahedra.

We prove existence and uniqueness of the discrete solution and

optimal order of convergence in the broken $H^1$-seminorm and

the $L^2$-norm.

The novelty of our approach is that

the usual integral compatibility condition of the discrete functions

across the element faces is modified such that it

can be solely treated on the

reference element once for all faces of the grid.

A numerical comparison between

conforming and nonconforming discretizations will be given in the

three-dimensional case.

**Keywords:**

nonconforming quadrilateral and hexahedral finite elements, multilevel grids, diffusion-convection-reaction equation, error estimates