In this talk we present results of application of the Finite Element Method (FEM) to mechanical problems across the length scales.

First, we will present the work related to matrix-free operators applied to finite-strain hyperelastic problems with geometric multigrid.

In order to improve the performance of iterative solvers within the high-performance computing context, so-called matrix-free method with sum factorization will be adopted. Different implementations of the finite-strain hyperelastic tangent operator are investigated numerically.

In order to improve the convergence behavior of iterative solvers, we also propose a method by which to construct level tangent operators and employ them to define a geometric multigrid preconditioner.

Next, we discuss atomistic-to-continuum coupling methods.

Such methods combine highly-accurate atomistic models in the region of interest (i.e. a crack tip) with numerically efficient continuum description to obtain computationally attractive solution approach.

Both staggered and concurrent coupling methods will be briefly discussed.

Finally, we will show results of application of the FEM to problems in Quantum Mechanics, namely the Density Functional Theory.

Advantages of the FEM as compared to standard discretization techniques (finite difference and plane waves) will be discussed,

including application of the hp-adaptive FEM and the Partition-of-Unity Method.

**Bio:**

Dr. Denis Davydov received his PhD in Engineering from the Czech Technical University in 2010. He is currently a research associate at the Chair of Applied Mechanics, Friedrich-Alexander University Erlangen-Nuremberg, Germany. He is interested in various topics within the realm of the Finite Element Method such as matrix-free approaches in solid mechanics, atomist-to-continuum coupling strategies, geometric multigrid preconditioners, partition of unity method and density functional theory.