arXiv subjects

Nis-Erik Bohne

Publications and source records attributed to Nis-Erik Bohne.

4 recordsLinked to original sources

arXiv2026

Discrete reliability for high-order Crouzeix--Raviart finite elements

In this paper, the adaptive numerical solution of a 2D Poisson model problem by Crouzeix-Raviart elements ($\operatorname*{CR}_{k}$ $\operatorname*{FEM}$) of arbitrary odd degree $k\geq1$ is investigated. The analysis is based on an established, abstract theoretical framework: the \textit{axioms of adaptivity} imply optimal convergence rates for the adaptive algorithm induced by a residual-type a posteriori error estimator. Here, we introduce the error estimator for the $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ discretization and our main theoretical result is the proof ot Axiom 3: \textit{discrete reliability}. This generalizes results for adaptive lowest order $\operatorname*{CR}_{1}$ $\operatorname*{FEM}$ in the literature. For this analysis, we introduce and analyze new local quasi-interpolation operators for $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ which are key for our proof of discrete reliability. We present the results of numerical experiments for the adaptive version of $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ for some low and higher (odd) degrees $k\geq1$ which illustrate the optimal convergence rates for all considered values of $k$.

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arXiv2024

Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions

In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $\left( d-1\right) $ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does \textbf{not} exist in higher space dimension and $k>1$.

math.NA↗

arXiv2024

The pressure-wired Stokes element: a mesh-robust version of the Scott-Vogelius element

The Scott-Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order $k$ and a discontinuous pressure approximation of order $k-1$. It employs a "singular distance" (measured by some geometric mesh quantity $ Θ\left( \mathbf{z}\right) \geq 0$ for triangle vertices $\mathbf{z}$) and imposes a local side condition on the pressure space associated to vertices $\mathbf{z}$ with $Θ\left( \mathbf{z}\right) =0$. The method is inf-sup stable for any fixed regular triangulation and $k\geq 4$. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices $0<Θ\left( \mathbf{z}\right) \ll 1$. In this paper, we introduce a very simple parameter-dependent modification of the Scott-Vogelius element such that the inf-sup constant is independent of nearly-singular vertices. We will show by analysis and also by numerical experiments that the effect on the divergence-free condition for the discrete velocity is negligibly small.

math.NA↗

arXiv2024

Pressure-improved Scott-Vogelius type elements

The Scott-Vogelius element is a popular finite element for the discretization of the Stokes equations which enjoys inf-sup stability and gives divergence-free velocity approximation. However, it is well known that the convergence rates for the discrete pressure deteriorate in the presence of certain $critical$ $vertices$ in a triangulation of the domain. Modifications of the Scott-Vogelius element such as the recently introduced pressure-wired Stokes element also suffer from this effect. In this paper we introduce a simple modification strategy for these pressure spaces that preserves the inf-sup stability while the pressure converges at an optimal rate.

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