## Structure, Regularity, and Robustness in the Approximation of Partial Differential Equations

### Motivation

Partial differential equations are ubiquitous in mathematical modeling, and so is the need of their numerical solution. When discretizing partial differential equations, their structure, the regularity of their solutions, and the robustness with respect to certain parameters play prominent roles.

The workshop intends to assess these roles, looking also at their interplay.

### Participation

Participation is free. If you plan to attend, please send an email to Pietro Zanotti not later than February 3, 2020.

### Schedule

Monday, February 10, 2020, 9:00-18:00
Sala di Rappresentanza del Dipartimento di Matematica, Via Saldini 50, 20133 Milano

09:00-09:05 Opening
09:05-09:50 R. Hiptmair - Discretizing the Advection of Differential Forms
09:55-10:40 E. Georgoulis - Hypocoercivity-compatible Finite Element Methods for Kinetic Equations

10:40-11:00 Coffee break in Aula C

12:35-14:30 Lunch break

16:05-16:25 Coffee break in Aula C

Andreas Veeser
Pietro Zanotti

### Support

 The workshop is supported by PRIN 2017 "Numerical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations (NA-FROM-PDEs)" Dipartimento di Matematica dell'Università degli Studi di Milano

### Abstracts

(in alphabetic order with respect to the speakers)

Some Comments on Structure Preserving High Order Schemes: the Example of Entropy Preserving Schemes, and Other Constraints
Rémi Abgrall (University of Zurich, Switzerland)

In this talk, I will consider the following question. Given a scheme designed for approximating an hyperbolic system of equation, how can it be modified so that an additional constraint is satisfied without destroying the accuracy. A typical example is fluid mechanics and the entropy. Another example, again in fluid mechanics, is, starting from a non conservative formulation of the problem, how can we build an approximation that will guaranty that the converged solution - if they exist - are weak solution of the problem ? If time permits, I will also consider the case of differential constraints.

An Introduction to Curved Virtual Elements
Lourenco Beirão da Veiga (University of Milano-Bicocca, Italy)

The Virtual Element Method (VEM) was introduced in [1,2] as a generalization of the Finite Element Method that allows for general polygonal and polyhedral meshes. Polytopal meshes can be very useful for a wide range of reasons, including meshing of the domain (such as cracks) and data (such as inclusions) features, automatic use of hanging nodes, moving meshes, adaptivity. By avoiding the explicit construction of the local basis functions, Virtual Elements can easily handle general polygons/polyhedrons without the need of an overly complex construction.

The scope of the present talk is to present Virtual Elements with curved faces, introduced in [3] and further developed in [4]. Indeed, all the VEM papers in the literature make use of polygonal and polyhedral meshes, i.e. with straight edges and faces. On the other hand, as acknowledged in the finite element (FEM) literature, especially for high order methods the approximation of the domain by facets introduces an error that can dominate the analysis. This issue has lead, for example, to the development of non affine isoparametric FEM elements and to Isogeometric Analysis.

In the context of Virtual Elements, one can exploit the peculiar construction of the method that (1) does not need an explicit expression of the basis functions and (2) is directly defined in physical space, i.e. no reference element is used. This allows to define discrete spaces also on elements that are curved in such a way to exactly represent the domain of interest. The needed ingredient is a (piecewise regular) parametrization of the boundary of the domain.

In the first part of the talk we present the curved VEM on a model elliptic problem, including the definition of the involved discrete space, the theoretical aspects and some numerical tests. In the second part, we introduce a variant of the discrete space that is more suitable for solid mechanics problems, and develop the method in the realm of small deformation elasticity and inelasticity, including a final set of numerical tests.

[1] L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23:199-214 (2013)
[2] L. Beirao da Veiga, F. Brezzi, L. D. Marini, A. Russo, The Hitchhiker’s Guide to the Virtual Element Method, Math. Models Methods Appl. Sci., 24:1541-1573 (2014)
[3] L. Beirao da Veiga, A. Russo, G. Vacca, The Virtual Element Method with curved edges, Math. Mod. Numer. Anal., 53:375-404 (2019)
[4] E. Artioli, L. Beirao da Veiga, F. Dassi, Curvilinear Virtual Elements for 2D solid mechanics applications, Comp. Meth. Appl. Mech. Engrg., DOI 10.1016/j.cma.2019.112667

Hypocoercivity-compatible Finite Element Methods for Kinetic Equations
Emmanuil Georgoulis (University of Leicester, UK, and Technical University of Athens, Greece)

We shall discuss some recent developments on a family of Galerkin finite element methods for kinetic equations with degenerate diffusion, such as the classical Kolmogorov and Fokker-Planck equations. In particular, diffusion is not present in all spatial directions. Nonetheless, the solutions to these problems admit typically decay properties to some long time equilibrium, depending on closure by suitable boundary/decay-at-infinity conditions. A key attribute of the proposed family of methods is that they also admit similar decay properties for very general families of triangulations. The method construction uses ideas by the general theory of hypocoercivity developed by Villani, along with judicious choice of numerical flux functions. These developments turn out to be sufficient to imply that the proposed finite element methods admit a priori error bounds with constants independent of the final time, despite these equations' degenerate diffusion nature. Thus, the new methods provably allow for robust error analysis for final times tending to infinity.

Discretizing the Advection of Differential Forms
Ralf Hiptmair (ETH Zurich, Switzerland)

Oscillation in A Posteriori Error Analysis
Christian Kreuzer (Technical University of Dortmund, Germany)

A posteriori error estimators are a key tool for the quality assessment of given finite element approximations to an unknown PDE solution as well as for the application of adaptive techniques.

Typically, the estimators are equivalent to the error up to an additive term, the so called oscillation. It is a common believe that this is the price for the `computability' of the estimator and that the oscillation is of higher order than the error. Cohen, DeVore, and Nochetto [CDN:2012], however, presented an example, where the error vanishes with the generic optimal rate, but the oscillation does not. Interestingly, in this example, the local $H^{-1}$-norms are assumed to be computed exactly and thus the computability of the estimator cannot be the reason for the asymptotic overestimation. In particular, this proves both believes wrong in general.

In this talk, we present a new approach to posteriori error analysis, where the oscillation is dominated by the error. The crucial step is a new splitting of the data into oscillation and oscillation free data. Moreover, the estimator is computable if the discrete linear system can essentially be assembled exactly.

[CDN:2012] A. Cohen, R. DeVore, and R. H. Nochetto, Convergence Rates of AFEM with H^{-1} Data, Found. Comput. Math. 12 (2012):671-718

This is a joint work with A. Veeser.

Gradient-robustness: a New Concept Assuring Accurate Spatial Discretizations for Vector-valued PDEs

Vector-valued PDEs like the incompressible Navier-Stokes equations (in primitive variables velocity and pressure) describe the time evolution of a vector-valued quantity like the momentum density. For vector-valued PDEs it is quite natural to derive formally on the continuous level a derived time evolution of the vorticity and the divergence. Accordingly, any space discretization for a vector-valued PDE (implicitly) delivers a discrete vorticity and discrete divergence equation.

While the celebrated inf-sup stability will be shown to assure an accurate discrete divergence equation, the talk will actually focus on the question, which structural properties allow for an implicitly defined accurate discrete vorticity equation. The key observation is that the L^2-orthogonality of divergence-free vector fields and gradient fields is the weak equivalent to the vector calculus identity $\nabla \times \nabla \psi = \mathbf{0}$ for arbitrary smooth scalar fields $\psi$.

In the context of the incompressible Navier-Stokes equations, the concept of pressure-robustness was introduced in 2016, in order to discriminate between space discretizations with accurate and inaccurate discrete vorticity equations. H(div)-conforming finite element spaces have been found as an important means to realize the L^2-orthogonality between discretely divergence-free vector fields and (arbitrary) gradient fields. Further, it is shown that spatial discretizations that are not pressure-robust may suffer from i) a degradation of the (preasymptotic) convergence rate, and ii) large, parameter-dependent constants in error estimates. Typical flows that benefit from pressure-robustness are quasi-hydrostatic flows, quasi-geostrophic flows and vortex-dominated high Reynolds number flows.

Last but not least, the talk shows how to extend the concept of pressure-robustness to more general vector-valued PDEs, leading to the concept of gradient-robustness. Gradient robustness assures that dominant and extreme gradient fields in a vector-valued PDE will not lead to inaccuracies in the discretization. The talk will show examples from linear elasticity and compressible (Navier-)Stokes flows at low Mach numbers and in stratified flows. Thus, connections to well-balanced schemes and WENO schemes will be drawn.

Structure Preserving Reduced Order Methods for Fluid-structure Interaction Parametric Problems: State of the Art and Perspectives
Gianluigi Rozza (SISSA, Trieste, Italy)

We describe the state of the art and perspectives in the developments of efficient structure preserving reduced order methods for parametric nonlinear fluid-structure interaction problems by monolithic and segregated approaches, as well as the use of numerical techniques to enhance the reduction of the Kolmogorov n-width in order to improve computational performances. Special attention is dedicated to the approximation stability of the reduced order model by supremisers, to the structure preserving property, as well as to the imposition of accurate and efficient coupling conditions to guarantee continuity of quantities at the fluid-structure interface.

Joint work with Monica Nonino, Francesco Ballarin (SISSA) and Yvon Maday (Sorbonne Paris LJLL).

A Fast Solver for High-degree IGA
Giancarlo Sangalli (University of Pavia, Italy)

The concept of k-refinement was proposed as one of the key features of isogeometric analysis, "a new, more efficient, higher-order concept", in the seminal work [1]. The idea of using high-degree and continuity splines/NURBS as a basis for a new high-order method appeared very promising from the beginning, and received confirmations from the next developments. The k-refinement leads to several advantages: higher accuracy per degree-of-freedom, improved spectral accuracy, the possibility of structure-preserving smooth discretizations are the most interesting features that have been studied actively in the community. At the same time, the k-refinement brings significant challenges at the computational level: using standard finite element routines, its computational cost grows with respect to the degree, making degree raising computationally expensive. This presentation gives an overview of some recent results that extend what we did in [2].

[1] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., Vol. 194, pp. 4135-4195 (2005).
[2] G. Sangalli and M. Tani, Matrix-free isogeometric analysis: the computationally efficient k-method, Comput. Methods Appl. Mech. Engrg., Vol. 338, pp. 117-133 (2018).

This research activity is developed with Monica Montardini, Mattia Tani, and other collaborators.

Last update: 22/01/2020 -- A. Veeser