An elegant theorem by J.L. Lions establishes well-posedness of non-autonomous evolutionary problems in Hilbert spaces which are defined by a non-autonomous form. However a regularity problem remained open for many years. We give a survey on positive and negative (partially very recent) results. One of the positive results can be applied to an evolutionary network which has been studied by Dominik Dier and Marjeta Kramar jointly with the speaker. It is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. Besides existence and uniqueness long-time behaviour can be described. When conductivity and diffusion coefficients match (so that mass is conserved) the solutions converge to an equilibrium.[-]

I will review (some of) the HPC solution strategies developed in Feel++. We present our advances in developing a language specific to partial differential equations embedded in C++. We have been developing the Feel++ framework (Finite Element method Embedded Language in C++) to the point where it allows to use a very wide range of Galerkin methods and advanced numerical methods such as domain decomposition methods including mortar and three fields methods, fictitious domain methods or certified reduced basis. We shall present an overview of the various ingredients as well as some illustrations. The ingredients include a very expressive embedded language, seamless interpolation, mesh adaption, seamless parallelisation. As to the illustrations, they exercise the versatility of the framework either by allowing the development and/or numerical verification of (new) mathematical methods or the development of large multi-physics applications - e.g. fluid-structure interaction using either an Arbitrary Lagrangian Eulerian formulation or a levelset based one; high field magnets modeling which involves electro-thermal, magnetostatics, mechanical and thermo-hydraulics model; ... - The range of users span from mechanical engineers in industry, physicists in complex fluids, computer scientists in biomedical applications to applied mathematicians thanks to the shared common mathematical embedded language hiding linear algebra and computer science complexities.[-]

Parametrized PDE (Partial Differential Equation) Apps are PDE solvers which satisfy stringent per-query performance requirements: less-than or approximate 5-second problem specification time; less-than or approximate 5-second problem solution time, field and outputs; less-than or approximate 5% solution error, specified metrics; less-than or approximate 5-second solution visualization time. Parametrized PDE apps are relevant in many-query, real-time, and interactive contexts such as design, parameter estimation, monitoring, and education.
In this talk we describe and demonstrate a PDE App computational methodology. The numerical approach comprises three ingredients: component => system synthesis, formulated as a static-condensation procedure; model order reduction, informed by evanescence arguments at component interfaces (port reduction) and low-dimensional parametric manifolds in component interiors (reduced basis techniques); and parallel computation, implemented in a cloud environment. We provide examples in acoustics and also linear elasticity.[-]

We consider an acoustic waveguide modeled as follows:

$ \left \{\begin {matrix}
\Delta u+k^2(1+V)u=0& in & \Omega= \mathbb{R} \times]0,1[\\
\frac{\partial u}{\partial y}=0& on & \partial \Omega
\end{matrix}\right.$

where $u$ denotes the complex valued pressure, k is the frequency and $V \in L^\infty(\Omega)$ is a compactly supported potential.
It is well-known that they may exist non trivial solutions $u$ in $L^2(\Omega)$, called trapped modes. Associated eigenvalues $\lambda = k^2$ are embedded in the essential spectrum $\mathbb{R}^+$. They can be computed as the real part of the complex spectrum of a non-self-adjoint eigenvalue problem, defined by using the so-called Perfectly Matched Layers (which consist in a complex dilation in the infinite direction) [1].
We show here that it is possible, by modifying in particular the parameters of the Perfectly Matched Layers, to define new complex spectra which include, in addition to trapped modes, frequencies where the potential $V$ is, in some sense, invisible to one incident wave.
Our approach allows to extend to higher dimension the results obtained in [2] on a 1D model problem.[-]

Recently there has been a surge in interest in cut, or unfitted, finite element methods. In this class of methods typically the computational mesh is independent of the geometry. Interfaces and boundaries are allowed to cut through the mesh in a very general fashion. Constraints on the boundaries such as boundary or transmission conditions are typically imposed weakly using Nitsche's method. In this talk we will discuss how these ideas can be combined in a fruitful way with the idea of hybridization, where additional degrees of freedom are added on the interfaces to further improve the decoupling of the systems, allowing for static condensation of interior unknowns. In the first part of the talk we will discuss how hybridization can be combined with the classical cut finite element method, using standard H1 -conforming finite elements in each subdomain, leading to a robust method allowing for the integration of polytopal geometries, where the subdomains are independent of the underlying mesh. This leads to a framework where it is easy to integrate multiscale features such as strongly varying coefficients, or multidimensional coupling, as in flow in fractured domains. Some examples of such applications will be given. In the second part of the talk we will focus on the Hybridized High Order Method (HHO) and show how cut techniques can be introduced in this context. The HHO is a recently introduced nonconforming method that allows for arbitrary order discretization of diffusive problems on polytopal meshes. HHO methods have hybrid unknowns, made of polynomials in the mesh elements and on the faces, without any continuity requirement. They rely on high-order local reconstructions, which are used to build consistent Galerkin contributions and appropriate stabilization terms designed to preserve the high-order approximation properties of the local reconstructions. Here we will show how cut element techniques can be introduced as a tool for the handling of (possibly curved) interfaces or boundaries that are allowed to cut through the polytopal mesh. In this context the cut element method plays the role of a local interface model, where the associated degrees of freedom are eliminated in the static condensation step. Issues of robustness and accuracy will be discussed and illustrated by some numerical examples.[-]

I will present an efficient implementation of the highly robust and scalable GenEO preconditioner in the high-performance PDE framework DUNE. The GenEO coarse space is constructed by combining low energy solutions of local generalised eigenproblems using a partition of unity. In this talk, both weak and strong scaling for the GenEO solver on over 15,000 cores will be demonstrated by solving an industrially motivated problem with over 200 million degrees of freedom in aerospace composites modelling. Further, it will be shown that for highly complex parameter distributions in certain real-world applications, established methods can become intractable while GenEO remains fully effective. In the context of multilevel Markov chain Monte Carlo (MLMCMC), the GenEO coarse space also plays an important role as an effective surrogate model in PDE-constrained Bayesian inference. The second part will therefore focus on the approximation properties of the GenEO coarse space and on a high-performance parallel implementation of MLMCMC.
This is joint work with Tim Dodwell (Exeter), Anne Reinarz (TU Munich) and Linus Seelinger (Heidelberg).[-]

Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test an angular space reduced order model for the linear radiative transfer equation based on reduced basis methods (RBMs). Our algorithm is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an upwind discontinuous Galerkin method for the physical space, with an efficient synthetic accelerated source iteration for the resulting linear system. Strategies are particularly proposed to tackle the challenges associated with the scattering operator within the RBM framework.
This is a joint work with Z.Peng, Y. Chen, and Y. Cheng.[-]