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.[-]

In domain decomposition methods, most of the computational cost lies in the successive solutions of the local problems in subdomains via forward-backward substitutions and in the orthogonalization of interface search directions. All these operations are performed, in the best case, via BLAS-1 or BLAS-2 routines which are inefficient on multicore systems with hierarchical memory. A way to improve the parallel efficiency of the method consists in working with several search directions, since multiple forward-backward substitutions and reorthogonalizations involve BLAS-3 routines. In the case of a problem with several right-hand-sides, using a block Krylov method is a straightforward way to work with multiple search directions. This will be illustrated with an application in electromagnetism using FETI-2LM method. For problems with a single right-hand-side, deriving several search directions that make sense from the optimal one constructed by the Krylov method is not so easy. The recently developed S-FETI method gives a very good approach that does not only improve parallel efficiency but can also reduce the global computational cost in the case of very heterogeneous problems.[-]

We present a lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic eld H on each edge, and the vertex values of the Lagrange multiplier p (used to enforce the solenoidality of the magnetic induction B = µH). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called ”first kind N´ed´elec” elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions. Hints on a whole family of elements will also be given.[-]

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.[-]

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.[-]

Ingrid Daubechies' construction of orthonormal wavelet bases with compact support published in 1988 started a general interest to employ these functions also for the numerical solution of partial differential equations (PDEs). Concentrating on linear elliptic and parabolic PDEs, I will start from theoretical topics such as the well-posedness of the problem in appropriate function spaces and regularity of solutions and will then address quality and optimality of approximations and related concepts from approximation the- ory. We will see that wavelet bases can serve as a basic ingredient, both for the theory as well as for algorithmic realizations. Particularly for situations where solutions exhibit singularities, wavelet concepts enable adaptive appproximations for which convergence and optimal algorithmic complexity can be established. I will describe corresponding implementations based on biorthogonal spline-wavelets.
Moreover, wavelet-related concepts have triggered new developments for efficiently solving complex systems of PDEs, as they arise from optimization problems with PDEs.[-]

Multigrid is an iterative method for solving large linear systems of equations whose Toeplitz system matrix is positive definite. One of the crucial steps of any Multigrid method is based on multivariate subdivision. We derive sufficient conditions for convergence and optimality of Multigrid in terms of trigonometric polynomials associated with the corresponding subdivision schemes.
(This is a joint work with Marco Donatelli, Lucia Romani and Valentina Turati).[-]