Lorenzo Mascotto

Virtual elements for Maxwell and MHD equations

 

We construct some virtual element spaces for the discretization of magnetic problems, including Maxwell equations and resistive magneto-hydrodynamic models. Such virtual elements are particularly suited for the approximation of magnetic problems, as they form exact sequences and satisfy exactly divergence-free constraints. Contextually, we review interpolation and stability properties of edge and face virtual elements. Simple numerical examples on manufactured solutions show that divergence-free constraints are not violated.

 

 

[Back to top]

Jeonghun Lee

TBA

 

[Back to top]

Snorre Christiansen

Finite Element Systems

 

The framework of Finite Element Systems is designed to provide a category theoretical foundation for finite element methods. I’ll discuss the framework and some examples.

 

[Back to top]

Duygu Sap

Discretization of BGG Complexes over Tensor-Product Meshes

 

In this talk, we present a discretization of the Bernstein-Gelfand-Gelfand (BGG) diagrams and complexes over tensor-product meshes in arbitrary dimensions. Our framework relies on the extension of one-dimensional piecewise polynomial spaces to n-dimensions. By utilizing the tensor-product nature of B-splines, we construct discretizations of the Hessian, elasticity, and div-div complexes via B-splines of arbitrary degree and regularity.

 

[Back to top]

Douglas Arnold

What the @#$! is cohomology doing in numerical analysis?!

 

TBA

 

[Back to top]

Ralf Hiptmair

Boundary Element Exterior Calculus

We consider first-kind boundary integral equations (BIEs) arising from boundary value problems for the Hodge-Laplacians and the Dirac operator associated with the de Rham complex on a bounded Lipschitz domain in Euclidean space. We discovered [1] that the variational formulations of these BIEs are those of the Hodge-Laplacians and Dirac operator belonging to the so-called trace de Rham complex [2], which is a Hilbert complex of differential forms on the boundary of the domain with non-local inner products. Hence the conforming boundary-element Galerkin discretization of these BIEs can rely on the framework of finite-element exterior calculus with the new twist of a non-local inner product. In particular, once we establish an h-uniform discrete Poincare inequality involving non-local norms, h-uniform stability of the boundary element method is settled.

[1] E. Schulz, R. Hiptmair, and S. Kurz, Boundary integral exterior calculus, Tech. Rep. 2022-36, Seminar for Applied Mathematics, ETH Zürich, 2022

[2] R. Hiptmair, D. Pauly, and E. Schulz, Traces for Hilbert complexes, J. Funct. Anal., 284 (2023)

 

[Back to top]

Rafael Vazquez

Fast computation of electromagnetic wave propagation with spline differential forms

 

We present an structure-preserving geometric method which exhibits high order of convergence and, contrarily to other high order geometric methods, does not rely on the geometric realization of any dual mesh. The method is based on the De Rham complex of tensor-product B-splines, from which we construct two exact sequences of differential forms: the primal sequence starts from splines of degree $p$ and at least $C^1$ continuity; the dual sequence starts from the space of tensor-product splines of degree $p-1$, which in the parametric domain coincides with the last space of the primal sequence. The differential operators (gradient, curl and divergence) are condensed in the exterior derivative operator. They are well defined for both sequences, and can be expressed in terms of incidence matrices associated to a Cartesian mesh. The method is completed with two sets of discrete Hodge-star operators relating the two sequences, mapping the space of primal k-forms into the space of dual (n-k)-forms, and vice versa. The discrete Hodge-star operators encapsulate all the metric-dependent properties, including material properties. We show a particular choice of the discrete Hodge-star operators inspired by [R. Hiptmair, Discrete Hodge operators, Numer Math. 90, 2001], and how to compute them through the fast inversion of Kronecker product matrices. We apply the method to the solution of Maxwell equations for electromagnetic wave propagation. The numerical tests confirm that the method conserves energy exactly, it has high order of convergence, and the computational cost is much lower than for standard Galerkin discretizations. We will also present preliminary results about the extension of the method to multi-patch geometries, going beyond the tensor-product structure.

 

[Back to top]

Kaibo Hu

Bounded Poincare maps for BGG complexes

 

Poincare integral operators give explicit potential and provide a null-homotopy inverse of differential operators. These operators play a key role in the mathematical and numerical analysis of fluid and electromagnetic problems. Consequences include the well-posedness of the Stokes problem and p-robustness of high order finite element methods. In this talk, we will review a generalized framework for Bernstein-Gelfand-Gelfand (BGG) complexes with weak regularity and derive bounded Poincare operators for these complexes with potential applications in elasticity and relativity. The idea is to carry over the results for the de Rham complex by Costabel and McIntosh to these cases by homological algebra. This is a joint work with Andreas Cap.

 

[Back to top]

Martin Campos Pinto

Bounded commuting projections for multipatch spaces

 

We present stable commuting projection operators on de Rham sequences of two-dimensional multipatch spaces with local tensor-product parametrization and non-matching interfaces. Our construction covers the case of shape-regular patches with different mappings and locally refined patches, under the assumption that neighbouring patches have nested resolutions and that interior vertices are shared by exactly four patches. Following a broken-FEEC approach we first apply a tensor-product construction on the single-patch de Rham sequences and modify the resulting patch-wise commuting projections to enforce their conformity while preserving their commuting, projection, and L2 stability properties. The resulting operators are local and stable in L2, with constants independent of both the size and the inner resolution of the individual patches.

 

 

[Back to top]

Theophile Chaumont-Frelet

Guaranteed and polynomial-degree-robust a posteriori estimates for mixed finite element discretizations of the curl-curl problem

 

In this talk, I will consider magnetostatic problems in mixed form, whereby the magnetic field H and magnetic potential A are respectively discretized with Nedelec and Raviart-Thomas finite elements. I will propose a post-processing strategy to reconstruct a divergence-conforming magnetic induction field B from the curl-conforming magnetic field H output by the finite element scheme. I will show that this reconstructed field B can be used to define an a posteriori error estimator that (a) delivers a constant-free upper bound on the error, (b) is locally efficient and (c) is polynomial-degree-robust, meaning that the efficiency constants do not depend on the polynomial degree. Crucially, the proposed construction do not rely on any specific assumption on the topology of the computational domain.

 

[Back to top]

Jeremias Arf

IGA for second-order Hilbert complexes

 

The fruitful connection between the concept of Finite Element Exterior Calculus (FEEC) and Isogeometric Analysis (IGA) has been demonstrated by the works of Buffa et al. in 2011 and the development of isogeometric differential forms. These advancements enabled the incorporation of curved geometries into numerical methods while preserving the exact sequence property of the discrete de Rham chains. Crucial to the definition of IGA spaces is the existence of suitable pullbacks, allowing the exploitation of the tensor product structure of B-splines in the reference domain. By using the “complexes from complexes” approach proposed by Arnold and Hu (2021), we show that suitable spline complexes can be defined for several Hilbert complexes involving second-order derivatives. However, when considering Hilbert complexes with higher-order differential operators, the task of finding compatible pullbacks that can also be used for NURBS geometries becomes challenging. Consequently, we focus on specific Hodge-Laplace problems where IGA ansatz spaces can be employed, and well-posedness can be demonstrated. Theoretical considerations are complemented by numerical tests, including applications in the field of linear elasticity. The results suggest that IGA can also be meaningfully utilized within the framework of higher-order complexes.

 

[Back to top]

Ludovico Bruni Bruno

Weights: from interpolation to FEEC

Weights are geometrical degrees of freedom for differential forms. They are defined as integral of $k$-forms over $k$-submanifolds of a given domain, say a simplex $ T $. This rather intuitive definition yields in turn a large class of interesting problems: how should these supports be chosen? Under whicht hypothesis they are unisolvent? How much interpolation properties are affected by their placement? In the introductory part of this talk, many of these questions will be properly formalised and answered, with a specific focus on unisolvence and symmetry. For these aspects differential geometrical- and algebraic-oriented techniques are involved. The aforementioned construction gives a generalisation of usual Lagrange interpolation: we present examples and drawbacks of this approach, including a Runge-like behaviour. As in the nodal case, a great flexibility in the choice of the supports allows to play with these quantities to improve interpolation properties. This is then used as the interpolator for finite elements method. Being a Galerkin method, the error in the approximate solution is unchanged as the finite dimensional space in which the solution is sought is not changed. In contrast, the matrix of the problem depends on the choice of these degrees of freedom, which might then be optimised in order to obtain better spectral properties. An example of this technique, along with a suitable strategy for the construction of a preconditioner, is presented.

 

[Back to top]

Jarle Sogn

IETI-DP methods for almost incompressible linear elasticity

 

We develop Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) methods for linear elasticity on multi-patch IgA domains. In resent years, IETI-DP and related methods have been studied extensively, mainly for the Poisson problem. For linear elasticity, several challenges arise such as material and geometry locking. To overcome material locking we introduce a mixed variational form, which has a saddle point structure. We derive IETI-DP methods and prove a convergence estimate which does not suffer from material locking and does not degenerate for certain geometries, like elongated rectangles.

 

[Back to top]

Constantin Greif

The Kolmogorov N-width for linear transport: Exact representation and the influence of the data

 

The Kolmogorov N-width describes the best possible error one can achieve by elements of an N-dimensional linear space. Particular interest has occurred within Model Order Reduction (MOR) of parameterized PDEs. While it is known that the N-width decays exponentially fast (and thus admits efficient MOR) for certain problems, there are examples of the linear transport and the wave equation, where the decay rate deteriorates to N^−1/2. We use techniques from Fourier Analysis to derive exact representations of the N-width in terms of initial and boundary conditions of the linear transport equation modeled by some function g for half-wave symmetric data. For arbitrary functions g, we derive bounds and prove that these bounds are sharp. In particular, we prove that the N-width decays as c_r N^−(r+1/2) for functions in the Sobolev space, g ∈ H^r. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds. To better preserve the structure of the solution, we propose a nonlinear MOR method that enhances the linear approximation by ridge functions, which also feature linear characteristics like the wave or transport equation.

 

[Back to top]

Francesca Bonizzoni

Discrete tensor product BGG sequences

 

Starting from the well-understood de Rham complex, it is possible to derive new complexes and deduce their properties from those of the starting complex. This construction is related to the Bernstein-Gelfand-Gelfand (BGG) sequence. The tensor product construction yields to tensor product versions of BGG sequences. In this talk we present a discretization of the BGG tensor product diagram by means of tensor product finite elements and splines on Cartesian meshes.

 

[Back to top]

Enrico Zampa

Multisymplecticity in finite element exterior calculus

 

Symplecticity is a fundamental property of Hamiltonian ODEs, and its preservation by numerical integrators is well studied. Hamiltonian PDEs, introduced by de Donder and Weyl, have an analogous property called multisymplecticity, which is expressed as a local conservation law satisfied by solutions. Recently, McLachlan and Stern (Found. Comput. Math., 2020) characterized the preservation of this property by a large family of hybridized conforming, nonconforming, mixed, and discontinuous Galerkin methods. In this work we extend their analysis to a more general class of multisymplectic PDEs introduced by Bridges (Proc. R. Soc. A., 2006), which includes semilinear Hodge–Dirac and Hodge–Laplace problems involving differential forms. More specifically, we show that such systems admit a multisymplectic conservation law, which is equivalent to the symmetry of the Dirichlet-to-Neumann operator for k-forms studied by Belishev and Sharafutdinov (Bull. Sci. Math., 2008). We then analyze several hybrid methods for the Hodge–Dirac problem, including the recently proposed hybridized version of Finite Element Exterior Calculus by Awanou, Fabien, Guzmán and Stern (Math. Comp., 2023), the extended Galerkin method by Hong, Li and Xu (Math. Comp., 2022), and the nonconforming method for the vector Laplacian proposed by Barker, Cao and Stern (2022). Joint work with Ari Stern (Washington University in St. Louis).

 

[Back to top]

Andrea Brugnoli

Dual field mixed and hybrid finite elements for port-Hamiltonian systems

 

In this talk, I will discuss a general overview hyperbolic port-Hamiltonian (pH) systems arising in electromagnetism and solid mechanics. The underlying geometric structure of pH systems, the Dirac structure, will be explained in a finite and infinite dimensional context. This general framework is then illustrated considering linear wave models arising in acoustic, electromagnetism and mechanics as well as general non linear elsatodynamics. In a second part of the talk the discretization is discussed. By using tools from FEEC, pH systems belonging to this class can be discretized in a structured way, ensuring the preservation of the power balance at the discrete level. Two approaches may be used to this aim: a mixed Galerkin formulation or an equivalent hybrid formulation that only solves for the globally coupled trace variables

 

[Back to top]

Simon Lemaire

Building bridges between polytopal methods

 

Within the past 20 years, a myriad of novel approaches for the numerical approximation of PDEs on general polytopal meshes have been introduced. Salient examples include the Hybridizable Discontinuous Galerkin (HDG) approach, the Virtual Element Method (VEM), the Hybrid High-Order (HHO) method, or the Discrete De Rham (DDR) paradigm. In this talk, we will review the main ideas underpinning such approaches, and build bridges between them, shedding light on their tight connections.

 

[Back to top]

Wietse Boon

Numerical methods for Stokes flow and Biot poroelasticity using rotation-based formulations

 

We propose a mixed finite element method for Stokes flow with one degree of freedom per element for the pressure and one per facet for the velocity on simplicial grids. The method is derived by first considering the vorticity–velocity–pressure formulation of the Stokes equations. The vorticity is then elimintated through the use of a local quadrature rule, leading to a multipoint vorticity mixed finite element method. The discrete solution possesses several advantageous properties. First, the velocity field is pointwise divergence-free and the method is pressure robust. Second, the application of the quadrature rule leaves several properties of the solution intact. For example, the pressure is invariant to the low-order integration on the vorticity variable. We derive theoretically that the method is linearly convergent in all variables and show second order convergence in the vorticity in special cases. Next, we remark on the similarities between the Stokes problem and Biot poroelasticity. Using an analogous approach as before, we introduce the solid rotation and fluid flux as auxiliary variables and subsequently removed from the system using a local quadrature rule. This leads to a multipoint rotation-flux mixed finite element method that employs the lowest-order Raviart–Thomas finite element space for the solid displacement and piecewise constants for the fluid pressure. Rigorous analysis shows the method to be linearly convergent and we obtain similar invariants to the quadrature rule. For both the Stokes and the Biot problems, the analytical results concerning stability, invariants, and convergence are confirmed by numerical experiments. To benchmark the methods, we consider a lid-driven cavity flow for the multipoint vorticity MFEM and Mandel’s problem for the multipoint rotation-flux MFEM.

 

[Back to top]

Markus Wess

High-order cell methods for time-dependent Maxwell equations

 

We are concerned with the efficient numerical approximation of solutions of the time-domain Maxwell system. Methods like the classical finite-difference time-domain method or finite integration techniques rely on the approximation of the electric and the magnetic field on two interlaced (Cartesian) grids respectively. Our method expands this idea to general triangular meshes by defining a dual grid using the barycentric subdivision of the primal mesh, resulting in a decomposition of each mesh into quadrilaterals. Contrary to previous approaches we use a Lagrangian polynomial basis with respect to tensor product integration points on the unit square which are subsequently re-mapped to the quadrilaterals of the reference element. This approach also provides a generalization of the method to three space dimensions and enables us to use lumped mass matrices which significantly increases the computational efficiency of our method. Numerical experiments underline the facts that the resulting algorithm provides converging, spurious-free solutions and is efficient, compared to competing methods.

 

[Back to top]

Yuwen Li

Nodal auxiliary space preconditioning for the surface de Rham complex

 

In this talk, I will present recent results about solving large-scale linear systems arising from the edge and face element discretizations on hypersurfaces. For those systems on unstructured triangulated surfaces, I will introduce uniform and user-friendly preconditioners motivated by the philosophy of nodal auxiliary space preconditioning. These preconditioners only require inverting several discrete surface Laplacians and lead to efficient iterative methods for computing harmonic tangential vector fields on discrete surfaces.

 

[Back to top]

Francesco Patrizi

Conforming/Non-Conforming Isogeometric de Rham complex discretization in disk-like domains via polar splines: applications to electromagnetism

 

In this talk we present a discretization of the continuous de Rham complex by means of adequate tensor-product spline spaces sustaining the same cohomological structure, when the underlying physical domain has a disk-like shape. Discretizations preserving such topological invariant of the physical model are commonly exploited in electromagnetics to obtain numerical solutions satisfying important conservation laws at the discrete level. Thereby one avoids spurious behaviors and, on the contrary, improves accuracy and stability of the approximations. The singularity of the parametrization of such physical domains demands the construction of suitable restricted spline spaces, called polar spline spaces, ensuring an acceptable smoothness to set up the discrete complex. In particular we present two sub-complexes, whose 0-forms are C^0 and C^1 smooth respectively, on the domain. Although it seems natural to adopt polar splines as basis functions, these do not have a tensor-product structure near the pole. This has a profound effect when pursuing high-performance computing: fast Kronecker-based algorithms cannot be used, MPI communication patterns are modified near the pole and new data structures may be needed. Therefore, in order to obtain a discretization capable to scale well with problems of large dimensions, we further present a, so called, conforming/non-conforming (CONGA) variation of the method. In this approach, the approximation is built in the ambient space of the tensor splines, by means of standard projectors which ease the parallelization of the processes, and afterwards such approximation is projected on the polar spline subspace to preserve the conformity of the discretization.

 

[Back to top]

Deepesh Toshniwal

Discretizing the de Rham complex using locally-refinable splines

 

Finite element exterior calculus (FEEC) is a framework for designing stable and accurate finite element discretizations for a wide variety of systems of PDEs. The involved finite element spaces are constructed using piecewise polynomial differential forms, and stability of the discrete problems is established by preserving at the discrete level the geometric, topological, algebraic and analytic structures that ensure well-posedness of the continuous problem. The framework achieves this using methods from differential geometry, algebraic topology, homological algebra and functional analysis. In this talk I will discuss the use of smooth splines within FEEC, they are the de facto standard for representing geometries of interest in engineering and offer superior accuracy in numerical simulations (per degree of freedom) compared to classical finite elements. In particular, I will discuss new results on the use of locally-refinable spline spaces for discretizing the de Rham complex.

 

[Back to top]

Adrian Ruf

Asymptotically preserving higher-order methods for nonlocal conservation laws

 

In this talk I will first give a brief overview of different ways to model nonlocal conservation laws. The focus will be on models that use a finite horizon to characterize nonlocal interactions inspired by the same notion used in peridynamics—a popular nonlocal alternative to classical elasticity theory with applications in material sciences. I will present new regularity results for a class of nonlocal conservation laws called the nonlocal pair-interaction model. These results motivate the design of higher-order numerical schemes which are asymptotically compatible with the underlying local conservation law. I will detail the construction of a second-order numerical scheme that generalizes the class of second-order reconstruction-based schemes for local conservation laws. It can be shown that the second-order scheme converges towards a weak solution, and—under certain assumptions on the nonlocal interaction kernel—even towards the unique entropy solution of the nonlocal conservation law. Such a result is currently out of reach for local conservation laws.

 

[Back to top]

Annalisa Buffa

Gluing spline patches for differential forms

 

TBA

 

[Back to top]

Joachim Schoeberl

Distributional Finite Elements with Applications in Elasticity and Curvature

 

Vector-valued function spaces, their finite element sub-spaces, and relations between these spaces are well understood within the de Rham complex. The framework of differential forms and Hilbert complexes provides a unified framework for any space dimension. Various matrix-valued finite element spaces have been introduced and analyzed more or less independently. In this presentation we put these spaces into a so called 2-complex.

 

[Back to top]

Ingeborg Gjerde

Network models for the flow of CSF in the brain

 

Our brain uses approximately 20% of the body’s energy in resting state. Yet, the brain lacks a traditional lymphatic system for metabolic clearance. This raises the question of how the brain clears metabolic waste. This question is of key importance as several neurological diseases, such as Alzheimer’s and Parkinson’s disease, are characterised by the accumulation of toxic proteins in the brain. According to the glymphatic theory, cerebrospinal fluid (CSF) flows in a network of perivascular spaces surrounding the arteries of the brain. This allows for a bulk flow of CSF through the brain that can remove metabolic waste. There are, however, several unanswered questions regarding the physical driving forces of this flow. In this talk, we present a Stokes type network model for simulating pulsatile glymphatic flow. We further introduce the concept of graph calculus and graph Sobolev spaces, and show how these can be used to formulate suitable discretization methods. With the numerical methods in hand, we show how network simulations can be set up to match experimental results, allowing us to probe the driving forces of CSF flow. In particular, our simulations show the importance of vasomotion associated with sleep.

 

[Back to top]

Tianwei Yu

Asymptotic-Preserving Discretization of Three-Dimensional Plasma Fluid Models

 

We elaborate a numerical method for a three-dimensional hydrodynamic multi-species plasma model which boils down to an extended Euler-Maxwell system. Our method is inspired by and extends the one-dimensional scheme from [P.~Degond, F.~Deluzet, and D.~Savelief, \emph{Numerical approximation of the Euler-Maxwell model in the quasineutral limit}, Journal of Computational Physics, 231 (4), pp.~1917–1946, 2012]. It can cope with large variations of the so-called Debye length λD and is robust in the quasi-neutral singular-perturbation limit λD→0, because it enjoys an \emph{asymptotic-preserving} (AP) property in the sense that in the sense that the limit λD→0 of the scheme yields a viable discretization for the continuous limit model. The key ingredients of our approach are (i) a discretization of Maxwell’s equations based on primal and dual meshes in the spirit of \emph{discrete exterior calculus} (DEC) also known the finite integration technique (FIT), (ii) a finite volume method (FVM) for the fluid equations on the dual mesh, (iii) \emph{mixed implicit-explicit timestepping}, (iv) special no-flux and contact boundary conditions at an outer cut-off boundary, and (v) additional \emph{stabilization} in the non-conducting region outside the plasma domain based on direct enforcement of Gauss’ law. Numerical tests provide strong evidence confirming the AP property of the new method.

 

[Back to top]

Wouter Tonnon

Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows

 

We develop a mesh-based semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions recast as a non-linear transport problem for a momentum 1-form. A linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to inviscid incompressible Euler flows. Conservation of energy and helicity are enforced separately.

 

[Back to top]