#### Prof. Dr. Fleurianne Bertrand

associated member

Numerical Mathematics

Humbolt University Berlin

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- fb@math.hu-berlin.de
- +30 2093 5855
- personal website

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**Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity**

Bertrand, F. and Kober, B. and Moldenhauer, M. and Starke, G.*Numerical Methods for Partial Differential Equations*(2021)This paper proposes and analyzes a posteriori error estimator based on stress equilibration for linear elasticity with emphasis on the behavior for (nearly) incompressible materials. It is based on an H(div)-conforming, weakly symmetric stress reconstruction from the displacement-pressure approximation computed with a stable finite element pair. Our focus is on the Taylor-Hood combination of continuous finite element spaces of polynomial degrees k + 1 and k for the displacement and the pressure, respectively. This weak symmetry allows us to prove that the resulting error estimator constitutes a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. It does not involve global constants like those from Korn's in equality which may become very large depending on the location and type of the boundary conditions. Local efficiency, also uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator. Numerical results for the popular Cook's membrane test problem confirm the theoretical predictions. © 2021 The Authors. Numerical Methods for Partial Differential Equations published by Wiley Periodicals LLC.view abstract 10.1002/num.22741 **Asymptotically Exact A Posteriori Error Analysis for the Mixed Laplace Eigenvalue Problem**

Bertrand, F. and Boffi, D. and Stenberg, R.*Computational Methods in Applied Mathematics*20 (2020)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. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.view abstract 10.1515/cmam-2019-0099 **Equilibrated Stress Reconstruction and a Posteriori Error Estimation for Linear Elasticity**

Bertrand, F. and Kober, B. and Moldenhauer, M. and Starke, G.*CISM International Centre for Mechanical Sciences, Courses and Lectures*597 (2020)Based on the displacement–pressure approximation computed with a stable finite element pair, a stress equilibration procedure for linear elasticity is proposed. Our focus is on the Taylor–Hood finite element space, with emphasis on the behavior for (nearly) incompressible materials. From a combination of displacement in the standard continuous finite element spaces of polynomial degrees k+1 and pressure in the standard continuous finite element spaces of polynomial degrees k, we construct an H(div)-conforming, weakly symmetric stress reconstruction. Explicit formulas are first given for a flux reconstruction and then for the stress reconstruction. © 2020, CISM International Centre for Mechanical Sciences.view abstract 10.1007/978-3-030-33520-5_3 **Weakly symmetric stress equilibration for hyperelastic material models**

Bertrand, F. and Moldenhauer, M. and Starke, G.*GAMM Mitteilungen*43 (2020)A stress equilibration procedure for hyperelastic material models is proposed and analyzed in this paper. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs, in a vertex-patch-wise manner, an H(div)-conforming approximation to the first Piola-Kirchhoff stress. This is done in such a way that its associated Cauchy stress is weakly symmetric in the sense that its antisymmetric part is zero tested against continuous piecewise linear functions. Our main result is the identification of the subspace of test functions perpendicular to the range of the local equilibration system on each patch which turn out to be rigid body modes associated with the current configuration. Momentum balance properties are investigated analytically and numerically and the resulting stress reconstruction is shown to provide improved results for surface traction forces by computational experiments. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheimview abstract 10.1002/gamm.202000007 **A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction**

Bertrand, F. and Moldenhauer, M. and Starke, G.*Computational Methods in Applied Mathematics*19 (2019)The nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement approximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable combination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction procedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin [2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect to the energy norm involving only known constants, two modifications are carried out: (i) the stress reconstruction in next-to-lowest order Raviart-Thomas spaces is modified in such a way that its anti-symmetric part vanishes in average on each element; (ii) the auxiliary conforming approximation is constructed under the constraint that its divergence coincides with the one for the nonconforming approximation. An important aspect of our construction is that all results hold uniformly in the incompressible limit. Global efficiency is also shown and the effectiveness is illustrated by adaptive computations involving different Lamé parameters including the incompressible limit case. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.view abstract 10.1515/cmam-2018-0004 **Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry**

Bertrand, F. and Cai, Z. and Park, E.Y.*Computational Methods in Applied Mathematics*19 (2019)This paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the L2 norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the norm of H(div; ω) for the stress, of H1(ω) for the displacement/velocity, and of L2(ω) for the rotation/vorticity and the pressure. This immediately implies optimal error estimates in the energy norm for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart-Thomas finite element spaces are used to approximate the stress tensor. Through a refined duality argument, an optimal L2 norm error estimates for the displacement/velocity are also established. Finally, numerical results for a Cook's membrane problem of planar elasticity are included in order to illustrate the robustness of our method in the incompressible limit. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.view abstract 10.1515/cmam-2018-0255 **Recent Advances in Least-Squares and Discontinuous Petrov-Galerkin Finite Element Methods**

Bertrand, F. and Demkowicz, L. and Gopalakrishnan, J. and Heuer, N.*Computational Methods in Applied Mathematics*19 (2019)Least-squares (LS) and discontinuous Petrov-Galerkin (DPG) finite element methods are an emerging methodology in the computational partial differential equations with unconditional stability and built-in a posteriori error control. This special issue represents the state of the art in minimal residual methods in the L2-norm for the LS schemes and in dual norm with broken test-functions in the DPG schemes. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.view abstract 10.1515/cmam-2019-0097 **An alternative proof of a strip estimate for first-order system least-squares for interface problems**

Bertrand, F.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*10665 LNCS (2018)The purpose of this paper is an alternative proof of a strip estimate, used in Least-Squares methods for interface problems, as in [4] for a two-phase flow problem with incompressible flow in the subdomains. The Stokes flow problems in the subdomains are treated as first-order systems and a combination of H(div) -conforming Raviart-Thomas and standard H1 -conforming elements were used for the discretization. The interface condition is built directly in the H(div) -conforming space. Using the strip estimate, the homogeneous Least-Squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates. © Springer International Publishing AG 2018.view abstract 10.1007/978-3-319-73441-5_9 **First-order system least-squares for interface problems**

Bertrand, F.*SIAM Journal on Numerical Analysis*56 (2018)The two-phase flow problem with incompressible flow in the subdomains is studied in this paper. The Stokes flow problems are treated as first-order systems, involving stress and velocity and using the L2 norm to define a least-squares functional. A combination of H(div)-conforming Raviart–Thomas and standard H1-conforming elements is used for the discretization. The interface conditions are directly in the H(div)-conforming finite element space. The homogeneous least-squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates. It also establishes the fact that the local evaluation of the least-squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies. © 2018 Society for Industrial and Applied Mathematics.view abstract 10.1137/16M1105827 **Parametric Raviart-Thomas elements for mixed methods on domains with curved surfaces**

Bertrand, F. and Starke, G.*SIAM Journal on Numerical Analysis*54 (2016)The finite element approximation on curved boundaries using parametric Raviart-Thomas spaces is studied in the context of the mixed formulation of Poisson's equation as a saddlepoint system. It is shown that optimal-order convergence is retained on domains with piecewise Ck+2 boundary for the parametric Raviart-Thomas space of degree k ≥ 0 under the usual regularity assumptions. This extends the analysis in [F. Bertrand, S. Münzenmaier, and G. Starke, SIAM J. Numer. Anal., 52 (2014), pp. 3165-3180] from the first-order system least squares formulation to mixed approaches of saddle-point type. In addition, a detailed proof of the crucial estimate in three dimensions is given which handles some complications not present in the two-dimensional case. Moreover, the appropriate treatment of inhomogeneous ux boundary conditions is discussed. The results are confirmed by computational results which also demonstrate that optimal-order convergence is not achieved, in general, if standard Raviart-Thomas elements are used instead of the parametric spaces. © 2016 Society for Industrial and Applied Mathematics.view abstract 10.1137/15M1045442 **First-order system least squares on curved boundaries: Higher-order Raviart-Thomas elements**

Bertrand, F. and Münzenmaier, S. and Starke, G.*SIAM Journal on Numerical Analysis*52 (2014)With this paper, our investigation of the finite element approximation on curved boundaries using Raviart-Thomas spaces in the context of first-order system least squares methods is continued and extended to the higher-order case. It is shown that the optimal order of convergence is retained from the lowest-order case if a parametric version of Raviart-Thomas elements is used. This is illustrated numerically for an elliptic boundary value problem involving a circular boundary curve. © 2014 Societ y for Industrial and Applied Mathematics.view abstract 10.1137/130948902 **First-order system least squares on curved boundaries: Lowest-order Raviart-Thomas elements**

Bertrand, F. and Münzenmaier, S. and Starke, G.*SIAM Journal on Numerical Analysis*52 (2014)The effect of interpolated edges of curved boundaries on Raviart-Thomas finite element approximations is studied in this paper in the context of first-order system least squares methods. In particular, it is shown that an optimal order of convergence is achieved for lowestorder elements on a polygonal domain. This is illustrated numerically for an elliptic boundary value problem involving circular curves. The computational results also show that a polygonal approximation is not sufficient to achieve convergence of optimal order in the higher-order case. © 2014 Society for Industrial and Applied Mathematics.view abstract 10.1137/13091720X

#### elasticity

#### finite element method

#### interface problems

#### numerical methods