## On error controlled numerical model reduction for finite element squared (FE2) procedures

### February 11 @ 12:00 pm - 1:00 pm

Fredrik Larsson is a professor in Structural Mechanics at Chalmers University of Technology in Sweden, where he also earned his PhD in 2003. He has worked on various problems in the field of Computational Mechanics and the development of finite element procedures in solid mechanics. Particular research interests are multiscale modeling, numerical model reduction and a posteriori error estimation.

The “Finite Element squared” (FE2) technique is a multiscale method for analyzing problems on two distinct length scales using the finite element method, typically a macroscale where the overall response is sought and a microscale where fine-scale features are resolved. Each macroscale quadrature point is connected to a (discretized) boundary value problem on a Representative Volume Element (RVE). For nonlinear and/or time-dependent problems, the nested problems must be solved concurrently, making the FE2 procedure is still computationally extremely demanding. However, the fact that many similar problems on RVEs are solved for with small number of input/output data makes the procedure well suited for reduction techniques applied to the discrete equations, here denoted Numerical Model Reduction. In order to obtain reliable approximations, it is of outmost importance to quantify, and control, the error associated with the reduction procedure.

In this contribution, we address a few different model problems, pertinent to heat flow, consolidation of porous media and non-linear elasticity. In particular, suitable error estimators are developed that (in the linear case) are guaranteed w.r.t. the full-fledged finite element solution. The key ingredients in the error estimator is the weak formulation of the RVE problem in space-time, construction of a suitable norm, and the definition of associated problem. The fact that the approximation is compared to the discrete finite element solution allows for explicit evaluation at low cost. Finally, the estimator is also extended to compute bounds on user-defined quantities of interest within the realm of goal-oriented error estimation.