Università degli Studi di Milano

June 24 and 25, 2021

Participation is free. If you plan to attend, please send an email to RANAPDE2021 not later than June 21, 2021.

09:00-09:05 Opening

09:05-09:50 L. Pavarino - *Block FETI-DP/BDDC preconditioners for mixed isogeometric discretizations of almost incompressible elasticity and applications*

09:55-10:40 D. Praetorius - *Goal-oriented adaptive finite element methods with optimal computational complexity*

10:40-11:00 Break

11:00-11:45 L. Beirão da Veiga - *An Introduction to Virtual Elements in 3D*

11:50-12:35 Martin Vohralík - *Polynomial-degree-robust a posteriori error estimation for the curl-curl problem*

09:05-09:50 L. Gastaldi - *Finite element approximation of a fictitious domain formulation for fluid-structure interactions*

09:55-10:40 M. Weiser - *Adaptive Spectral Deferred Correction Methods from long time to parallel in time integration*

10:40-11:00 Break

11:00-11:45 Th. Wihler - *Gradient flow FEM with energy-based adaptivity for the Gross-Pitaevskii equation*

11:50-12:35 P. Antonietti - *Numerical models for earthquake ground motion and seismic risk assessment*

12:35-12:40 Closing

**Numerical models for earthquake ground motion and seismic risk assessment**

*Paola Antonietti (Politecnico di Milano, Italy)*

Physics-based numerical simulations of earthquake ground motion can be used to better understand the physics of earthquakes, improve the design of site-specific structures, and evaluate the seismic hazard, a key step for the reliable assessment of seismic risk.

The distinguishing features of a numerical method designed for seismic wave propagation are: accuracy, geometric flexibility and parallel scalability. High-order methods ensure low dissipation and dispersion errors. Geometric flexibility allows for complicated geometries and sharp discontinuities in the mechanical properties. Finally, since earthquake models are typically posed on domains that are very large compared to the wavelengths of interest, scalability allows to efficiently solve the resulting algebraic systems featuring several millions of unknowns.

In this talk we present an overview on numerical modelling of the ground motion induced by seismic waves based on employing high-order Discontinuous Galerkin methods. We also present two new approaches to couple the ground motion induced by earthquakes with the induced structural damages of buildings. The first one is based on empirical laws (fragility curves) whereas the second one employs physics-based linear and non-linear differential models. To validate the first approach we study the seismic damages in the Beijing area as a consequence of ground motion scenarios with magnitude in the range 6.5-7.3 Mw. To validate the second approach we consider the 1999 Mw6 Athens earthquake and study the seismic response of the Acropolis hill and of the Parthenon. Our numerical results have been obtained using the open-source numerical code SPEED.

** An Introduction to Virtual Elements in 3D**

*Lourenco Beirão da Veiga (Università di Milano-Bicocca, Italy)*

The Virtual Element Method (VEM), is a very recent technology introduced in 2013 for the discretization of partial differential equations, that has shared a good success in recent years. The VEM can be interpreted as a generalization of the Finite Element Method that allows general polygonal and polyhedral meshes, still keeping the same coding complexity and allowing for arbitrary degree of accuracy. In addition to the possibility to handle general polytopal meshes, the flexibility of the above construction yields other interesting properties with respect to more standard Galerkin methods (for instance, the VEM easily allows to build discrete spaces of arbitrary C^k regularity, or to satisfy exactly the divergence-free constraint for incompressible fluids).

The present talk is an introduction to the VEM, aiming at showing the main ideas of the method. We consider for simplicity a simple elliptic model problem (that is pure diffusion with variable coefficients) but set ourselves in the more involved 3D setting. In the first part we introduce the adopted Virtual Element space and the associated degrees of freedom, first by addressing the faces of the polyhedrons (i.e. polygons) and then building the space in the full volumes. We then describe the construction of the discrete bilinear form and the ensuing discretization of the problem. Furthermore, we show a set of theoretical and numerical results. In the very final part, we will give a glance at more involved problems, such as magnetostatics (mixed problem with more complex spaces interacting), (Navier-)Stokes equations and large deformation elasticity (nonlinear problem).

**Finite element approximation of a fictitious domain formulation
for fluid-structure interactions**

*Lucia Gastaldi (Università di Brescia, Italy)*

In this talk we describe a computational model for the simulation of fluid-structure interaction problems based on a fictitious domain approach. Our formulation originated from the Immersed Boundary Method and then moved toward Fictitious Domain approach. The finite element discretization of this formulation leads to an unfitted scheme. Our formulation for FSI offers an alternative to body-fitted approaches avoiding the difficulties related with mesh generation. Moreover, it allows the treatment of fluid and solid in their natural Eulerian and Lagrangian framework.

We present the well-posedness of our formulation at the continuous level in a simplified setting. We describe various time semi-discretizations that provide unconditionally stable schemes and discuss the choice of the finite element spaces to be used in order to have a stable and convergent scheme.

**Block FETI-DP/BDDC preconditioners for mixed isogeometric discretizations
of almost incompressible elasticity and applications**

*Luca Pavarino (Università di Pavia, Italy)*

FETI-DP (Finite Element Tearing and Interconnecting Dual-Primal) and BDDC (Balancing Domain Decomposition by Constraints) are two very successful domain decomposition algorithms for a variety of elliptic problems. We introduce parallel deluxe FETI-DP/BDDC preconditioners for almost incompressible elasticity and Stokes problems discretized by mixed isogeometric analysis with continuous pressures. The convergence rate depends on the model inf-sup constant and the condition number of a closely related BDDC algorithm for compressible elasticity. This bound is scalable in the number of subdomains, polylogarithmic in the ratio of subdomain and element sizes, and robust with respect to material incompressibility and parameters discontinuities across subdomain interfaces. We will conclude by reviewing ongoing work extending FETI-DP/BDDC preconditioners to cardiac electromechanical models and integrating them into cardiac finite element libraries.

O. B. Widlund, S. Zampini, S. Scacchi and L. F. Pavarino, Block FETI-DP/BDDC preconditioners for mixed isogeometric discretizations of three-dimensional almost incompressible elasticity. Math. Comp. 90, pp. 773--1797, 2021.

**Goal-oriented adaptive finite element methods with optimal computational complexity**

*Dirk Praetorius (Technische Universität Wien, Austria)*

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design a goal-oriented adaptive finite element method (GOAFEM), which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method (PCG). We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates with respect to the number of degrees of freedom as well as the computational cost.

The talk is based on joint work with Roland Becker (U Pau, FR), Gregor Gantner (U Amsterdam, NL), and Michael Innerberger (TU Wien, AT).

**Polynomial-degree-robust a posteriori error estimation for the curl-curl problem**

*Martin Vohralík (INRIA, Paris, France)*

We derive two types of a posteriori error estimates for Nédélec finite element discretizations of the curl-curl problem. In the first case, we proceed by a “broken patchwise equlibration” relying on smaller edge-based patches and related to the localization of the residual with test functions in H_0(curl) on the patches. In the second case, we design a patchwise equilibration relying on larger vertex-based patches and related to the localization of the residual with test functions in H(curl) on the patches and the partition-of-unity functions weighting the residual. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and constant-free in the second case. Stable extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge/vertex are a central theoretical tool. Practically, one has to construct a H(curl)-conforming Nédélec piecewise polynomial with the curl prescribed by a suitable Raviart-Thomas piecewise polynomial. This is structurally more difficult in the second case, where it is related to a divergence-free decomposition of a given divergence-free H(div)-conforming Raviart-Thomas piecewise polynomial. Numerical results illustrate the theoretical developments.

This is a joint work with Alexandre Ern (first case) and with Théophile Chaumont-Frelet (both cases).

**Adaptive Spectral Deferred Correction Methods from long time to parallel in time integration**

*Martin Weiser (Zuse Institut Berlin, Germany)*

Spectral deferred correction (SDC) methods can be interpreted as particular iterative solvers for implicit Runge-Kutta methods of collocation type, which are based on sweeps of low-order schemes, typically Euler steps. As such, they offer great flexibility in preconditioning and inexact computation.

In this talk, we will survey some of these options. In particular we will investigate the generalization from simple Euler sweeps to diagonally implicit Runge-Kutta sweeps and its implication on convergence rate, convergence theory for inexact sweeps with application to long-time integration of implant wear, spatially restricted sweeps as a simple multi-rate time stepping in cardiac electrophysiology, and compressed communication in hybrid para-real schemes for parallel in time integration. The examples will be illustrated by numerical examples.

**Gradient flow FEM with energy-based adaptivity for the Gross-Pitaevskii equation**

*Thomas Wihler (Universität Bern, Switzerland)*

We present an effective adaptive procedure for the numerical approximation of the steady-state Gross-Pitaevskii equation. Our approach is solely based on energy minimization, and consists of a combination of a novel adaptive finite element mesh refinement technique, which does not rely on any a posteriori error estimates, and a recently proposed new gradient flow. Numerical tests show that this strategy is able to provide highly accurate results, with optimal convergence rates with respect to the number of degrees of freedom.

Last update: 07/06/2021 -- A. Veeser