Rolf Stenberg - Academia.edu (original) (raw)

Papers by Rolf Stenberg

SIAM Journal on Scientific Computing, 2020

We show quasi-optimality and a posteriori error estimates for the frictionless contact problem be... more We show quasi-optimality and a posteriori error estimates for the frictionless contact problem between two elastic bodies with a zero-gap function. The analysis is based on interpreting Nitsche's method as a stabilised finite element method for which the error estimates can be obtained with minimal regularity assumptions and without the saturation assumption. We present three different Nitsche's mortaring techniques for the contact boundary each corresponding to a different stabilising term. Our numerical experiments show the robustness of Nitsche's method and corroborates the efficiency of the a posteriori error estimators.

We consider the Mixed Interpolated (Tensorial Components) finite element families for the Reissne... more We consider the Mixed Interpolated (Tensorial Components) finite element families for the Reissner-Mindlin plate model. For the case of a convex domain with clamped boundary conditions we prove regularity results and derive new error estimates which are uniformly valid with respect to the thickness parameter.

SIAM Journal on Scientific Computing, 2021

We introduce a Nitsche's method for the numerical approximation of the Kirchhoff-Love plate equat... more We introduce a Nitsche's method for the numerical approximation of the Kirchhoff-Love plate equation under general Robin-type boundary conditions. We analyze the method by presenting a priori and a posteriori error estimates in mesh-dependent norms. Several numerical examples are given to validate the approach and demonstrate its properties.

For the model Poisson problem we propose a method combining the discontinuous Galerkin method wit... more For the model Poisson problem we propose a method combining the discontinuous Galerkin method with a mixed formulation. In the method independent and fully discontinuous basis functions are used both for the scalar unknown and its flux. The continuity requirement is imposed by Nitsche's technique [7]. In the implementation the flux is eliminated by local condensing. We show that the method is stable and optimally convergent for all positive values of the stability parameter. We also perform an a posteriori error analysis. The theoretical results are verified by numerical computations.

This report contains the program, list of participants, and abstracts for the invited presentatio... more This report contains the program, list of participants, and abstracts for the invited presentations of the international conference Perspectives in Numerical Analysis 2008, held at the Helsinki University of Technology, May 27–29, 2008. AMS subject classifications: 65-06

Portugaliae Mathematica, 2019

We derive optimal a priori and a posteriori error estimates for Nitsche's method applied to unila... more We derive optimal a priori and a posteriori error estimates for Nitsche's method applied to unilateral contact problems. Our analysis is based on the interpretation of Nitsche's method as a stabilised finite element method for the mixed Lagrange multiplier formulation of the contact problem wherein the Lagrange multiplier has been eliminated elementwise. To simplify the presentation, we focus on the scalar Signorini problem and outline only the proofs of the main results since most of the auxiliary results can be traced to our previous works on the numerical approximation of variational inequalities. We end the paper by presenting results of our numerical computations which corroborate the efficiency and reliability of the a posteriori estimators. * Funding from Tekes (Decision number 3305/31/2015) and the Finnish Cultural Foundation is gratefully acknowledged.

Applied Mathematical Modelling, 2015

We derive a novel and rigorous correction to the classical Reynolds lubrication approximation for... more We derive a novel and rigorous correction to the classical Reynolds lubrication approximation for fluids with viscosity depending upon the pressure. Our analysis shows that the pressure dependence of viscosity leads to additional nonlinear terms related to the shear-rate and arising from a non negligible cross-film pressure. We present a numerical comparison between the classical Reynolds equation and our modified equation and conclude that the modified equation leads to the prediction of higher pressures and viscosities in the flow domain.

Numerische Mathematik, 1988

A new mixed finite element formulation for the equations of linear elasticity is considered. In t... more A new mixed finite element formulation for the equations of linear elasticity is considered. In the formulation the variables approximated are the displacement, the unsymmetric stress tensor and the rotation. The rotation act as a Lagrange multiplier introduced in order to enforce the symmetry of the stress tensor. Based on this formulation a new family of both twoand three-dimensional mixed methods is defined. Optimal error estimates, which are valid uniformly with respect to the Poisson ratio, are derived. Finally, a new postprocessing scheme for improving the displacement is introduced and analyzed.

ESAIM: Mathematical Modelling and Numerical Analysis, 2003

In this note, we propose and analyse a method for handling interfaces between nonmatching grids b... more In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.

Computer Methods in Applied Mechanics and Engineering, 1995

We introduce a new triangular element for nearly incompressible elasticity and incompressible flu... more We introduce a new triangular element for nearly incompressible elasticity and incompressible fluid flow. The method consists of conforming linear elements for one of the displacement (or velocity for flows) component and linear non-conforming elements for the other component. The element is proved to give an optimal approximation and this is also confirmed by several numerical examples.

BIT Numerical Mathematics, 2008

... Markus Hegland, Andreas Hellander, and Per Lötstedt consider the curse of dimensionality conn... more ... Markus Hegland, Andreas Hellander, and Per Lötstedt consider the curse of dimensionality connected with numerical solution of the chemical master equa-Page 2. 164 Perspectives in Numerical Analysis 2008 tion when the number of species grows. ...

International Journal for Numerical Methods in Engineering, 2009

This paper introduces and analyses a local, residual based a posteriori error indicator for the M... more This paper introduces and analyses a local, residual based a posteriori error indicator for the Morley finite element method of the biharmonic Kirchhoff plate bending problem. In the theoretical part of the paper, a recent approach presented by the authors for clamped boundaries is extended to general boundary conditions. The error indicator is proven to be both reliable and efficient. The numerical part of the paper presents a set of results on various benchmark computations with different kinds of domains and boundary conditions. These tests verify the reliability and efficiency of the error estimator and illustrate the robustness of the method for adaptive mesh refinements.

ArXiv, 2021

We consider mixed finite element methods with exact symmetric stress tensors. We derive a new qua... more We consider mixed finite element methods with exact symmetric stress tensors. We derive a new quasi-optimal a priori error estimate uniformly valid with respect to the compressibility. For the a posteriori error analysis we consider the Prager-Synge hypercircle principle and introduce a new estimate uniformly valid in the incompressible limit. All estimates are validated by numerical examples.

We survey the Nitsche's master-slave finite element method for elastic contact problems analy... more We survey the Nitsche's master-slave finite element method for elastic contact problems analysed in [2]. The main steps of the error analysis are recalled and numerical benchmark computations are presented.

SIAM Journal on Scientific Computing

We derive a residual a posteriori estimator for the Kirchhoff plate bending problem. We consider ... more We derive a residual a posteriori estimator for the Kirchhoff plate bending problem. We consider the problem with a combination of clamped, simply supported, and free boundary conditions subject to both distributed and concentrated (point and line) loads. Extensive numerical computations are presented to verify the functionality of the estimators.

Numerische Mathematik

In a previous paper [6] we have extended Nitsche's method [8] for the Poisson equation with gener... more In a previous paper [6] we have extended Nitsche's method [8] for the Poisson equation with general Robin boundary conditions. The analysis required that the solution is in H s , with s > 3/2. Here we give an improved error analysis using a technique proposed by Gudi [5].

Computational Methods in Applied Mathematics

This paper derives a posteriori error estimates for the mixed numerical approximation of the Lapl... more This paper derives a posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem. We discuss a reconstruction in the standard {H_{0}^{1}} -conforming space for the primal variable of the mixed Laplace eigenvalue problem and compare it with analogous approaches present in the literature for the corresponding source problem. In the case of Raviart–Thomas finite elements of arbitrary polynomial degree, the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. Our reconstruction is performed locally on a set of vertex patches.

Rakenteiden Mekaniikka

We outline the results of our recent article on the a posteriori error analysis of C1 finite elem... more We outline the results of our recent article on the a posteriori error analysis of C1 finite elements for the classical Kirchhoff plate model with general boundary conditions. Numerical examples are given.