Natural Sciences
Life Sciences
Scientific Computing
Natural Sciences

Rolf Stenberg, Department of Mathematics and Systems Analysis, Aalto University, Finland

Mathematikon, Conference Room, 5th Floor, Im Neuenheimer Feld 205

Interdisciplinary Center for Scientific Computing (IWR)

IWR Colloquium We survey our recent and ongoing work [1,2] on finite element methods for contact problems. Our approach is to first write the problem in mixed form, in which the contact pressure act as a Lagrange multiplier. In order to avoid the problems related to a direct mixed finite element discretisation, we use a stabilised formulation, in which appropriately weighted residual terms are added to the discrete variational forms. We prove that the formulation is uniformly stable, which implies an optimal a priori error estimate. Using the stability of the continuous problem, we also prove a posteriori estimates, the optimality of which is ensured by local lower bounds. In the implementation of the methods, the discrete Lagrange multiplier is locally eliminated, leading to a Nitsche-type method [3]. For the problems of a membrane and plate subject to solid obstacles, we present numerical results. Joint work with Tom Gustafsson (Aalto) and Juha Videman (Lisbon). References: [1] T. Gustafsson, R. Stenberg, J. Videman. Mixed and stabilized finite element methods for the obstacle problem. SIAM Journal of Numerical Analysis 55 (2017) 2718–2744 [2] T. Gustafsson, R. Stenberg, J. Videman. Stabilized methods for the plate obstacle problem. [3] E. Burman, P. Hansbo, M.G. Larson, R. Stenberg. Galerkin least squares finite element method for the obstacle problem. Computer Methods in Applied Mechanics and Engineering 313 (2017) 362–374

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