DFG SPP 1748/1: "Reliable Simulation Techniques in Solid Mechanics  Development of NonStandard Discretisation Methods, Mechanical and Mathematical Analysis"  Participating Project: "Foundation and Application of Generalized Mixed FEM Towards Nonlinear Problems in Solid Mechanics"
The research of this project aims at the mathematical foundation and the engineering application of generalized mixed FEM as well as the development and the analysis of new nonstandard methods that yield guaranteed results for nonlinear problems in solid mechanics. The practical applications in computational engineering will be the focus of the Workgroup LUH at the Leibniz University Hannover in cooperation with the Workgroup HU at the Humboldt Universität zu Berlin with focus on mathematical foundation of the novel discretization schemes. The joint target is the effective and reliable simulation in nonlinear continuum mechanics with development of adaptive numerical discretizations based on ultraweak formulations between nonconforming, mixed and discontinuous Galerkin or PetrovGalerkin Finite Element Methods. In the first funding period, the workgroup LUH developed different discontinuous discretization methods. An efficient extension/enhancement of the original discontinuous Galerkin Finite Element Method (dG FEM) avoids shearlocking effects and volumetriclocking for (nearly) incompressible and elastoplastic material behaviour. The workgroup HU developed and analysed a discontinuous PetrovGalerkin (dPG) FEM for a nonlinear model problem in collaboration with the workgroup LUH and proved optimal convergence rates of adaptive dPG and leastsquares methods for linear elastic problems. Further topics of research were guaranteed error bounds for pointwise symmetric discretizations in linear elasticity and the analysis of nonconforming FEM for polyconvex materials.
Principal Investigators
Carstensen, Carsten Prof. (Details) (Numerical Handling of Differential Equations) 

Wriggers, Peter Prof. Dr.Ing. habil. Dr. h.c. mult. Dr.Ing. E. h. (Leibniz University Hannover) 

Duration of Project
Start date: 09/2014
End date: 11/2019
Publications
P. Bringmann and C. Carstensen. An adaptive leastsquares FEM for the Stokes equations with optimal convergence rates. Numer. Math., 135(2):459–492, 2017.
P. Bringmann and C. Carstensen. hadaptive leastsquares finite element methods for the 2D Stokes equations of any order with optimal convergence rates. Comput. Math. Appl., 74(8):1923–1939, 2017.
P. Bringmann, C. Carstensen, D. Gallistl, F. Hellwig, D. Peterseim, S. Puttkammer, H. Rabus, and J. Storn. Towards adaptive discontinuous PetrovGalerkin methods. PAMM. Proc. Appl. Math. Mech., 16(1):741–742, 2016.
P. Bringmann, C. Carstensen, and G. Starke. An adaptive leastsquares FEM for linear elasticity with optimal convergence rates. SIAM J. Numer. Anal., 56(1):428–447, 2018.
C. Carstensen, P. Bringmann, F. Hellwig, and P. Wriggers. Nonlinear discontinuous petrovgalerkin methods. Numer. Math., 139(3):529–561, 2018.
C. Carstensen, L. Demkowicz, and J. Gopalakrishnan. Breaking spaces and forms for the DPG method and applications including Maxwell equations. Comput. Math. Appl., 72(3):494–522, 2016.
C. Carstensen and F. Hellwig. Loworder discontinuous PetrovGalerkin finite element methods for linear elasticity. SIAM J. Numer. Anal., 54(6):3388–3410, 2016.
C. Carstensen and F. Hellwig. Optimal convergence rates for adaptive lowestorderdiscontinuous PetrovGalerkin schemes. SIAM J. Numer. Anal., 56(2):1091–1111, 2018.
C. Carstensen, E. J. Park, and P. Bringmann. Convergence of natural adaptive least squares FEMs. Numer. Math., 136(4):1097–1115, 2017.
C. Carstensen, D. Peterseim, and A. Schröder. The norm of a discretized gradient in H(div)* for a posteriori finite element error analysis. Numer. Math., 132:519–539, 2016.
C. Carstensen and H. Rabus. Axioms of adaptivity with separate marking for data resolution. SIAM J. Numer. Anal., 55(6):2644–2665, 2017.
C. Carstensen and S. Puttkammer. A loworder discontinuous PetrovGalerkin method for the stokes equations. Numer. Math., 140(1):1–34, 2018.
C. Carstensen and J. Storn. Asymptotic exactness of the leastsquares finite element residual. SIAM J. Numer. Anal., 56(4):2008–2028, 2018.
T. Steiner and P. Wriggers. A primal discontinuous PetrovGalerkin finite element method for linear elasticity. PAMM. Proc. Appl. Math. Mech., 17(1):83–86, 2017.
T. Steiner, P. Wriggers, and S. Löhnert. A discontinuous galerkin finite element method for linear elasticity using a mixed integration scheme to circumvent shearlocking. PAMM. Proc. Appl. Math. Mech., 16:769–770, 2016.