Mira Schedensack

Comparison Results of Finite Element Methods for the Poisson Model Problem

This paper establishes the equivalence of the conforming Courant finite element method, the nonconforming Crouzeix--Raviart finite element method, and several first-order discontinuous Galerkin finite element methods in the sense that the respective energy error norms are equivalent up to generic constants and higher-order data oscillations in a Poisson model problem. The Raviart--Thomas mixed finite element method is better than the previous methods, whereas the conjecture of the converse relation is proved to be false. This paper completes the analysis of comparison initiated by Braess [Calcolo, 46 (2009), pp. 149--155]. Two numerical benchmarks illustrate the comparison theorems and the possible strict superiority of the Raviart--Thomas mixed finite element method. Applications include least-squares finite element methods, finite volume methods, and equality of approximation classes for concepts of optimality for adaptive finite element methods.

Quasi-optimal adaptive pseudostress approximation of the Stokes Equations

The pseudostress-velocity formulation of the stationary Stokes problem allows some quasi-optimal Raviart--Thomas mixed finite element formulation for any polynomial degree. The adaptive algorithm employs standard residual-based explicit a posteriori error estimation from Carstensen, Kim, and Park [SIAM J. Numer. Anal., 49 (2011), pp. 2501--2523] for the lowest-order Raviart--Thomas finite element functions in a simply connected Lipschitz domain. This paper proves optimal convergence rates in terms of the number of unknowns of the adaptive mesh-refining algorithm based on the concept of approximation classes. The proofs use some novel equivalence to first-order nonconforming Crouzeix--Raviart discretization plus a particular Helmholtz decomposition of deviatoric tensors.

Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems

The nonconforming approximation of eigenvalues is of high practical interest because it allows for guaranteed upper and lower eigenvalue bounds and for a convenient computation via a consistent diagonal mass matrix in 2D. The first main result is a comparison which states equivalence of the error of the nonconforming eigenvalue approximation with its best-approximation error and its error in a conforming computation on the same mesh. The second main result is optimality of an adaptive algorithm for the effective eigenvalue computation for the Laplace operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of a nonlinear approximation class. The analysis includes an inexact algebraic eigenvalue computation on each level of the adaptive algorithm which requires an iterative algorithm and a controlled termination criterion. The analysis is carried out for the first eigenvalue in a Laplace eigenvalue model problem in 2D.

Discrete Reliability for Crouzeix-Raviart FEMs

The proof of optimal convergence rates of adaptive finite element methods relies on Stevensons concept of discrete reliability. This paper proves the general discrete reliability for the nonconforming Crouzeix--Raviart finite element method on multiply connected domains in any space dimension. A novel discrete quasi-interpolation operator of first-order approximation involves an intermediate triangulation and acts as the identity on unrefined simplices, to circumvent any Helmholtz decomposition. Besides the generalization of the known application to any dimension and multiply connected domains, this paper outlines the optimality proof for uniformly convex minimization problems. This discrete reliability implies reliability for the explicit residual-based a posteriori error estimator in any space dimension and for multiply connected domains.

A New Generalization of the P 1 Non-Conforming FEM to Higher Polynomial Degrees

Abstract This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

A New Discretization for

This paper introduces new mixed formulations and discretizations for

m

m

th-Laplace Equations with Arbitrary Polynomial Degrees

th-Laplace equations of the form

Relaxing the CFL Condition for the Wave Equation on Adaptive Meshes

(-1)^m\Delta^m u=f

Thermo-optical interactions in a dye-microcavity photon Bose–Einstein condensate

for arbitrary