Dietmar Gallistl

Guaranteed lower eigenvalue bounds for the biharmonic equation

The computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate’s vibrations. This paper shows that the eigenvalue provided by the nonconforming Morley finite element analysis, which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense, is not always a lower bound. A fully-explicit error analysis of the Morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound. This paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower-order terms.

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.

An optimal adaptive FEM for eigenvalue clusters

The analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods. A first analysis for multiple eigenvalues of the recent work Dai et al. (arXiv Preprint 1210.1846v2, 2013) introduces an adaptive method whose marking strategy is based on the element-wise sum of local error estimator contributions for multiple eigenvalues. This paper proves the optimality of a practical adaptive algorithm based on a lowest-order conforming finite element method for eigenvalue clusters for the eigenvalues of the Laplace operator in terms of nonlinear approximation classes. All estimates are explicit in the initial mesh-size, the eigenvalues and the cluster width to clarify the dependence of the involved constants.

Multiscale Petrov-Galerkin Method for High-Frequency Heterogeneous Helmholtz Equations

This paper presents a multiscale Petrov-Galerkin finite element method for time-harmonic acoustic scattering problems with heterogeneous coefficients in the high-frequency regime. We show that the method is pollution-free also in the case of heterogeneous media provided that the stability bound of the continuous problem grows at most polynomially with the wave number k. By generalizing classical estimates of Melenk (Ph.D. Thesis, 1995) and Hetmaniuk (Commun. Math. Sci. 5, 2007) for homogeneous medium, we show that this assumption of polynomially wave number growth holds true for a particular class of smooth heterogeneous material coefficients. Further, we present numerical examples to verify our stability estimates and implement an example in the wider class of discontinuous coefficients to show computational applicability beyond our limited class of coefficients.

Low-order dPG-FEM for an elliptic PDE

This paper introduces a novel lowest-order discontinuous PetrovGalerkin (dPG) finite element method (FEM) for the Poisson model problem. The ultra-weak formulation allows for piecewise constant and affine ansatz functions and for piecewise affine and lowest-order RaviartThomas test functions. This lowest-order discretization for the Poisson model problem allows for a direct proof of the discrete infsup condition and a complete apriori and aposteriori error analysis. Numerical experiments investigate the performance of the method and underline the quasi-optimal convergence.

A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes

The discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh-refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it. One approach for this is the careful design of an intermediate triangulation with one-level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation. This paper enfolds the second possibility of some novel discrete Helmholtz decomposition for the nonconforming Morley finite element method. This guarantees the optimality of a standard adaptive mesh-refining algorithm for the biharmonic equation. Numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre-asymptotic range of the adaptive Morley finite element method.

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 posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.

Adaptive Nonconforming Finite Element Approximation of Eigenvalue Clusters

Abstract This paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system.