arXiv subjects

Carsten Carstensen

Publications and source records attributed to Carsten Carstensen.

36 records · Page 2Linked to original sources

arXiv2022

Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian

An extra-stabilised Morley finite element method (FEM) directly computes guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplace Dirichlet eigenvalues. The smallness assumption $\min\{λ_h,λ\}h_{\max}^{4}$ $\le 184.9570$ in $2$D (resp. $\le 21.2912$ in $3$D) on the maximal mesh-size $h_{\max}$ makes the computed $k$-th discrete eigenvalue $λ_h\le λ$ a lower eigenvalue bound for the $k$-th Dirichlet eigenvalue $λ$. This holds for multiple and clusters of eigenvalues and serves for the localisation of the bi-Laplacian Dirichlet eigenvalues in particular for coarse meshes. The analysis requires interpolation error estimates for the Morley FEM with explicit constants in any space dimension $n\ge 2$, which are of independent interest. The convergence analysis in $3$D follows the Babuška-Osborn theory and relies on a companion operator for the Morley finite element method. This is based on the Worsey-Farin $3$D version of the Hsieh-Clough-Tocher macro element with a careful selection of center points in a further decomposition of each tetrahedron into $12$ sub-tetrahedra. Numerical experiments in $2$D support the optimal convergence rates of the extra-stabilised Morley FEM and suggest an adaptive algorithm with optimal empirical convergence rates.

math.NA↗

arXiv2022

Lowest-order equivalent nonstandard finite element methods for biharmonic plates

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the $C^0$ interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side $F\in H^{-2}(Ω)$ replaced by $F\circ (JI_{\rm M}) $ and then are quasi-optimal in their respective discrete norms. The smoother $JI_{\rm M}$ is defined for a piecewise smooth input function by a (generalized) Morley interpolation $I_{\rm M}$ followed by a companion operator $J$. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj: Comparison results of nonstandard $P_2$ finite element methods for the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends the work [Veeser, Zanotti: Quasi-optimal nonconforming methods for symmetric elliptic problems, SIAM J. Numer. Anal. 56 (2018)] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.

math.NA↗

arXiv2022

Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the $m$-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix-Raviart ($m=1$) or Morley ($m=2$) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated $L^2$ error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in $3$D and highlights a new version of discrete reliability.

math.NA↗

arXiv2021

A priori and a posteriori error analysis of the Crouzeix-Raviart and Morley FEM with original and modified righthand sides

This article on nonconforming schemes for $m$ harmonic problems simultaneously treats the Crouzeix-Raviart ($m=1$) and the Morley finite elements ($m=2$) for the original and for modified right-hand side $F$ in the dual space $V^*:=H^{-m}(Ω)$ to the energy space $V:=H^{m}_0(Ω)$. The smoother $J:V_{\rm nc} \to V$ in this paper is a companion operator, that is a linear and bounded right-inverse to the nonconforming interpolation operator $I_{\rm nc} :V\to V_{\rm nc}$, and modifies the discrete right-hand side $F_h:=F\circ J \in V_{\rm nc}^*$. The best-approximation property of the modified scheme from Veeser et al. (2018) is recovered and complemented with an analysis of the convergence rates in weaker Sobolev norms. Examples with oscillating data show that the original method may fail to enjoy the best-approximation property but can also be better than the modified scheme. The a~posteriori analysis of this paper concerns data oscillations of various types in a class of right-hand sides $F\in V^*$. The reliable error estimates involve explicit constants and can be recommended for explicit error control of the piecewise energy norm. The efficiency follows solely up to data oscillations and examples illustrate this can be problematic.

math.NA↗

arXiv2020

Adaptive Morley FEM for the von Kármán equations with optimal convergence rates

The adaptive nonconforming Morley finite element method (FEM) approximates a regular solution to the von Kármán equations with optimal convergence rates for sufficiently fine triangulations and small bulk parameter in the Dörfler marking. This follows from the general axiomatic framework with the key arguments of stability, reduction, discrete reliability, and quasiorthogonality of an explicit residual-based error estimator. Particular attention is on the nonlinearity and the piecewise Sobolev embeddings required in the resulting trilinear form in the weak formulation of the nonconforming discretisation. The discrete reliability follows with a conforming companion for the discrete Morley functions from the medius analysis. The quasiorthogonality also relies on a novel piecewise $H^1$ a~priori error estimate and a careful analysis of the nonlinearity.

math.NA↗

arXiv2020

Morley Finite Element Method for the von Kármán Obstacle Problem

This paper focusses on the von Kármán equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Kármán obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von Kármán obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate.

math.NA↗

arXiv2019

Nonconforming Finite Element Discretisation for Semilinear Problems with Trilinear Nonlinearity

The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of 2D Navier-Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a~priori error analysis and an a~posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data and so has to provide discrete stability by a perturbation analysis before the Newton-Kantorovic theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix-Raviart FEM for second-order linear non-selfadjoint and indefinite elliptic problems with $L^\infty$ coefficients. The paper identifies six parameters and sufficient conditions for the local a~priori and a~posteriori error control of conforming and nonconforming discretisations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a~posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.

math.NA↗

arXiv2019

How to prove the discrete reliability for nonconforming finite element methods

Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity. One key ingredient is the discrete reliability of a residual-based a posteriori error estimator, which controls the error of two discrete finite element solutions based on two nested triangulations. In the error analysis of nonconforming finite element methods, like the Crouzeix-Raviart or Morley finite element schemes, the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper. The nonconforming interpolation operator, which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet, allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition. The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation. The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices. Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.

math.NA↗

arXiv2017

Nonlinear discontinuous Petrov-Galerkin methods

The discontinuous Petrov-Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper. Its lowest-order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation, a reduced formulation, and a weighted nonlinear least-squares method. Quasi-optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dPG framework for the approximation of a regular solution. The variational model example allows for a built-in guaranteed error control despite inexact solve. The subtle uniqueness of discrete minimizers is monitored in numerical examples.

math.NA↗

arXiv2017

Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-Interpolation

This paper provides a discrete Poincaré inequality in $n$ space dimensions on a simplex $K$ with explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero on $K$ and all integrals of jumps zero along all interior sides by its Lebesgue norm by $C(n)\operatorname{diam}(K)$. The explicit constant $C(n)$ depends only on the dimension $n=2,3$ in case of an adaptive triangulation with the newest vertex bisection. The second part of this paper proves the stability of an enrichment operator, which leads to the stability and approximation of a (discrete) quasi-interpolator applied in the proofs of the discrete Friedrichs inequality and discrete reliability estimate with explicit bounds on the constants in terms of the minimal angle $ω_0$ in the triangulation. The analysis allows the bound of two constants $Λ_1$ and $Λ_3$ in the axioms of adaptivity for the practical choice of the bulk parameter with guaranteed optimal convergence rates.

math.NA↗

arXiv2017

A Priori and A Posteriori Error Control of Discontinuous Galerkin Finite Element Methods for the Von Kármán Equations

This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Kármán equations defined on a polygonal domain. A discrete inf-sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established and this allows the proof of local existence and uniqueness of a discrete solution to the non-linear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel $C^0$ interior penalty method (IPDG). Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known $C^0$-IPDG lead to complications with some non-residual type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a non convex domain.

math.NA↗

arXiv2017

Interaction of martensitic microstructures in adjacent grains

It is often observed that martensitic microstructures in adjacent polycrystal grains are related. For example, micrographs of Arlt exhibit propagation of layered structures across grain boundaries in the cubic-to-tetragonal phase transformation in $\rm BaTiO_3$. Such observations are related to requirements of compatibility of the deformation at the grain boundary. Using a generalization of the Hadamard jump condition, this is explored in the nonlinear elasticity model of martensitic transformations for the case of a bicrystal with suitably oriented columnar geometry, in which the microstructure in both grains is assumed to involve just two martensitic variants, with a planar or non-planar interface between the grains.

cond-mat.mtrl-sci↗

arXiv2016

Axioms of adaptivity for separate marking

Mixed finite element methods with flux errors in $H(div)$-norms and div-least-squares finite element methods require a separate marking strategy in obligatory adaptive mesh-refining. The refinement indicator $σ^2(\mathcal T,K)=η^2(\mathcal T,K)+μ^2(K)$ of a finite element domain $K$ in an admissible triangulation $\mathcal T$ consists of some residual-based error estimator $η(\mathcal T,K)$ with some reduction property under local mesh-refining and some data approximation error $μ(K)$. Separate marking means either Dörfler marking if $μ^2(\mathcal T) \leq κη^2(\mathcal T)$ or otherwise an optimal data approximation algorithm runs with controlled accuracy as established in [Carstensen, Rabus, Math.Comp. 2011; Rabus, J.Numer.Math. 2015]. The axioms are abstract and sufficient conditions on the estimators $η(\mathcal T,K)$ and data approximation errors $μ(K)$ for optimal asymptotic convergence rates. The enfolded set of axioms simplifies \cite{CFP14} for collective marking, treats separate marking established for the first time in an abstract framework, generalizes [Carstensen, Park, SIAM J.Numer.Anal. 2015] for least-squares schemes, and extends [Carstensen, Rabus, Math.Comp. 2011] to the mixed FEM with flux error control in $H(div)$.

math.NA↗

arXiv2015

Geometry of polycrystals and microstructure

We investigate the geometry of polycrystals, showing that for polycrystals formed of convex grains the interior grains are polyhedral, while for polycrystals with general grain geometry the set of triple points is small. Then we investigate possible martensitic morphologies resulting from intergrain contact. For cubic-to-tetragonal transformations we show that homogeneous zero-energy microstructures matching a pure dilatation on a grain boundary necessarily involve more than four deformation gradients. We discuss the relevance of this result for observations of microstructures involving second and third-order laminates in various materials. Finally we consider the more specialized situation of bicrystals formed from materials having two martensitic energy wells (such as for orthorhombic to monoclinic transformations), but without any restrictions on the possible microstructure, showing how a generalization of the Hadamard jump condition can be applied at the intergrain boundary to show that a pure phase in either grain is impossible at minimum energy.

cond-mat.mtrl-sci↗

arXiv2014

Comparison results for the Stokes equations

This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P1 non-conforming FEM. The main comparison result is that the error of the P2-P0-FEM is a lower bound to the error of the Bernardi-Raugel (or reduced P2-P0) FEM, which is a lower bound to the error of the P1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs. Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition.

math.NA↗

arXiv2014

Error analysis of nonconforming and mixed FEMs for second-order linear non-selfadjoint and indefinite elliptic problems

The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite element discretization which converges owing to some a priori $L^2$ error estimates even for reduced regularity on non-convex polygonal domains. An equivalence result of that nonconforming finite element scheme to the mixed finite element method (MFEM) leads to the well-posedness of the discrete solution and to a priori error estimates for the MFEM. The explicit residual-based a posteriori error analysis allows some reliable and efficient error control and motivates some adaptive discretization which improves the empirical convergence rates in three computational benchmarks.

math.NA↗

arXiv2013

Axioms of Adaptivity

This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error estimators. Solely four axioms guarantee the optimality in terms of the error estimators. Compared to the state of the art in the temporary literature, the improvements of this article can be summarized as follows: First, a general framework is presented which covers the existing literature on optimality of adaptive schemes. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator. In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant. Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the $R$-linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007. Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

math.NA↗

arXiv2011

Uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods

Assumed stress hybrid methods are known to improve the performance of standard displacement-based finite elements and are widely used in computational mechanics. The methods are based on the Hellinger-Reissner variational principle for the displacement and stress variables. This work analyzes two existing 4-node hybrid stress quadrilateral elements due to Pian and Sumihara [Int. J. Numer. Meth. Engng, 1984] and due to Xie and Zhou [Int. J. Numer. Meth. Engng, 2004], which behave robustly in numerical benchmark tests. For the finite elements, the isoparametric bilinear interpolation is used for the displacement approximation, while different piecewise-independent 5-parameter modes are employed for the stress approximation. We show that the two schemes are free from Poisson-locking, in the sense that the error bound in the a priori estimate is independent of the relevant Lame constant $λ$. We also establish the equivalence of the methods to two assumed enhanced strain schemes. Finally, we derive reliable and efficient residual-based a posteriori error estimators for the stress in $L^{2}$-norm and the displacement in $H^{1}$-norm, and verify the theoretical results by some numerical experiments.

math.NA↗