Crouzeix-Raviart type finite elements on anisotropic meshes

Abstract

The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges. A numerical test is described.Mathematics Subject Classification (1991): 65N30; 65N15, 65N50, 65D05

Discontinuous Least-Squares finite element method for the Div-Curl problem

“…Moreover, we use an isotropic mesh refinement algorithm, splitting each element marked for refinement into two to eight new elements by successively dividing the longest edge, see [7] for more details. For these singular problems, it is however clear that we would gain from using an anisotropic error estimator and mesh refinement, see for instance [2], [20], and [29].…”

Anisotropic nonconforming Crouzeix-Raviart type FEM for second-order elliptic problems

“…For the nonconforming Crouzeix-Raviart type linear triangular element, up to now, only the convergence analysis on anisotropic right triangular meshes was researched in [10,17]. Obviously, anisotropic right triangular mesh subdivision restricts its serviceable range.…”

“…As for the low-order Crouzeix-Raviart type rectangular nonconforming finite elements, Apel et al [10] presented a modified one with the asymmetric shape function span {1, x, y, x 2 } or span {1, x, y, y 2 } and derived the optimal-order error estimates on anisotropic meshes in which the longer edges must parallel to the x-axis or y-axis. In order to overcome this shortcoming, Lin et al [11] and Shi et al [12] investigated a rectangular Crouzeix-Raviart type nonconforming element with the symmetric shape function span{1, x, y, x 2 , y 2 } on regularity meshes and anisotropic meshes, respectively.…”

“…As a first attempt, we try to fill this gap for second-order elliptic problems. By using of the anisotropic interpolation theorem(see Lemma 2.1 below) and introducing auxiliary finite element space similar to [10], the optimal-order error estimates of the broken energy norm and L 2 -norm are obtained with careful analysis when the maximal angle condition and coordinate system condition proposed in [3] are satisfied, although the analysis is more difficult than [10].…”

Nonconforming H 1-Galerkin mixed FEM for Sobolev equations on anisotropic meshes

“…But when the domain concerned is very narrow, if we employ the regular partition, the computing cost will be very high. The obvious idea to overcome this difficulty is to use the anisotropic meshes with fewer degrees of freedom [1][2][3]6,7,17,18] .…”