Schedule for: 24w5218 - Contemporary Challenges in Trefftz Methods, from Theory to Applications

Opening of the Workshop Contemporary Challenges in Trefftz Methods, from Theory to Applications, by Ilaria Pergia and Lise-Marie Imbert-Gérard.

Starting from recent ideas by Parolin-Huybrechs-Moiola, we study the approximation properties of propagative and non propagative plane waves in various domains. If the domain is a square, our finding is that one can define new families which are Hilbertian with respect to the D-trace scalar product on the boundary (or the N-trace scalar product on the boundary). It opens the possibility to improve on the condition number of the matrices generated by Trefftz methods.
This is a collaboration with Nikola Galante (PhD) and Emile Parolin.

Trefftz Discontinuous Galerkin (DG) methods provide a way to reduce the computational costs of DG methods. Recently, with the introduction of embedded, weak and quasi-Trefftz DG methods, the range of applications for the Trefftz DG paradigm has increased. In this talk we apply the embedded Trefftz methodology for solving the Stokes equations as an example for a vectorial PDE.
Discrete solutions of a corresponding method fulfill the Stokes equation pointwise within each element and yield element-wise divergence-free solutions, but are not normal-continuous. Due to the Trefftz ansatz velocity and pressure unknowns are strongly coupled on an element level. This gives rise to a special structure in the discrete Stokes saddle-point problem. We explain the structure and outline a full a-priori error analysis. Further, implementational aspects for the construction of Trefftz bases and the handling of inhomogeneous r.h.s. forcings are discussed and a comparison with other non-conforming methods is sketched. Finally, we present numerical examples and discuss current limitations and possible extensions.

We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz-type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non-constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. As typical for Trefftz-DG methods, we observe a severe reduction of the globally coupled unknowns when compared to standard discontinuous Galerkin methods, reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves.

The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well-known that it can be very effective assuming regularity of the domain and boundary conditions. The main drawback of the MFS is that the matrices involved are typically ill-conditioned and this may prevent the method from achieving high accuracy.
In this work, we propose a new algorithm to remove the ill-conditioning of the classical MFS in the context of the Laplace equation defined in planar domains. The main idea is to expand the MFS basis functions in terms of harmonic polynomials. Then, using the singular value decomposition and Arnoldi orthogonalization we define well conditioned basis functions spanning the same functional space as the MFS's. Several numerical examples show that when possible to be applied, this approach is much superior to previous approaches, such as the classical MFS or the MFS-QR.

Plane-wave Trefftz methods for the Helmholtz equation offer significant advantages over standard discretization approaches. However, a disadvantage of these methods which cannot be overlooked is the inherent poor conditioning in the resulting system matrices. We carefully examine the conditioning of the plane-wave discontinuous Galerkin method with respect to a single physical element where the properties of the mass and stiffness matrices depend on the size and geometry of the domain. We begin the analysis by considering a single disk-shaped element. In this setting, the condition of the matrices can be quantified in a straightforward way. In addition, the matrices related to these elements have characteristics that allow for easily constructed preconditioners. This is followed by numerical investigation of simple preconditioning strategies on more practical polygonal domains where we employ preconditioners with characteristics similar to the matrices for the disk-shaped element.
This is joint work with Nilima Nigam at Simon Fraser University.

NGSolve is a versatile finite element solver that offers efficient numerical treatment of partial differential equations. The software's computational core, coded in C++, ensures high performance, while its flexible Python interface enhances user accessibility. We will give an overview of some of NGSolve's functionality, underscoring its practicality and adaptability for researchers. In a hands-on demonstration, we will walk through the process of setting up and solving a simple PDE problem using the Python interface. We will also discuss some of the advanced features of NGSolve, and show how to work with the C++ code.

One of the prototypical applications of Trefftz methods is the approximation of solutions to the Helmholtz equation $\Delta u+k^2u=0$, with positive wavenumber $k$, by propagative plane waves (PPWs) $\mathbf{x}\mapsto e^{ik\mathbf{x}\cdot\mathbf{d}}$. However the representation of Helmholtz solutions by linear combinations of PPWs is notoriously unstable: to approximate some smooth solutions, linear combinations of PPWs require huge coefficients. In computer arithmetics, this leads to numerical cancellation and prevents any accuracy. This can be shown rigorously for the unit ball in 2D and 3D. This instability is often described in terms of ill-conditioning.
A remedy to such instability is the use of evanescent plane waves (EPWs): plane waves with complex propagation vectors $\mathbf{d}$. We show that any Helmholtz solution $u$ on a ball can be written as a continuous superposition of EPWs, and that the coefficient density is bounded by the $H^1$ norm of $u$. We propose a discretization strategy of this representation, based on modern sampling techniques, to approximate any solution $u$ with a finite combination of EPWs with bounded coefficients. The theory is supported by numerical experiments on the ball and on convex shapes, including the application to Trefftz discontinuous Galerkin (TDG) schemes.
This is a joint work with Nicola Galante (INRIA Paris), Daan Huybrechs (KU Leuven), and Emile Parolin (INRIA Paris).

We present a space-time ultra-weak discontinuous Galerkin discretization of the linear time-dependent Schrödinger equation. We prove that the method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. We discuss four different choices of discrete spaces: (i) a non-polynomial Trefftz space of complex wave functions, (ii) the full polynomial space, (iii) a quasi-Trefftz polynomial space, and (iv) a polynomial Trefftz space.
Several numerical experiments validate the accuracy and advantages of the proposed method.

The aim of this work is focused on a novel Trefftz method to approximate accurately the solution of time-harmonic wave motion problems in acoustics and structural dynamics involving layered materials with frequency-dependent physical properties, particularly at middle and high frequencies. Classical finite element methods (FEM) based on polynomials (even at high-order) are computationally intensive and suffer from phase leaks and pollution phenomena at high-frequency regimes. The proposed numerical method is a modal-based partition of unity finite element method (PUFEM), which utilizes a set of closed-form eigenfunctions as part of the modal basis for a related auxiliary time-harmonic wave motion problem, which can be computed off-line, usually without taking into account all the complexities of the geometrical and physical information of the original time-harmonic wave propagation problem. This combination of a model basis and the partition of unity allows for an accurate representation of the solution at the middle and high-frequency contributions at a reduced computational cost. The method is particularly efficient for problems with a known modal basis in simple geometries and homogeneous isotropic materials. Some numerical results are shown to illustrate the robustness of the proposed method. Finally, an industrial application of this numerical methodology will be used to detect cracks in layered materials will be highlighted.

Trefftz schemes are high-order Galerkin methods whose discrete functions are elementwise exact solutions of the underlying PDE. Since a family of local exact solutions is needed, Trefftz basis functions are usually restricted to PDEs that are linear, homogeneous and with piecewise-constant coefficients. If the equation has varying coefficients construction of suitable discrete Trefftz spaces is usually of reach. Quasi-Trefftz methods have been introduced to overcome this limitation, relying on discrete functions that are elementwise “approximate solutions” of the PDE, in the sense of Taylor polynomials. The main advantage of Trefftz and quasi-Trefftz schemes over more classical ones is the higher accuracy for comparable numbers of degrees of freedom.
In this talk, we present polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients, describe their optimal approximation properties and provide a simple algorithm to compute the basis functions, based on the Taylor expansion of the PDE’s coefficients. Then, we focus on a quasi-Trefftz DG method for the diffusion-advection-reaction equation with varying coefficients, showing stability and high-order convergence of the scheme. We also extend the method to non-homogeneous problems with piecewise-smooth source term, constructing a local quasi-Trefftz particular solution and then solving for the difference. We present numerical experiments in 2 and 3 space dimensions that show excellent properties in terms of approximation and convergence rate.
This is joint work with Lise-Marie Imbert-Gérard, Andrea Moiola and Paul Stocker.
References:
   [1] L.M. Imbert-Gérard, A. Moiola, C. Perinati and P. Stocker, A quasi-Trefftz DG method for the diffusion-advection-reaction equation with piecewise-smooth coefficients, in preparation.
   [2] C. Perinati, A quasi-Trefftz discontinuous Galerkin method for the homogeneous diffusion-advection-reaction equation with piecewise-smooth coefficients, Master’s thesis, University of Pavia, 2023. arXiv preprint arXiv:2312.09919.

Variational Trefftz methods are discontinuous Galerkin methods whose basis functions are local solutions of the PDE under consideration. In the context of homogeneous problems, analytical solutions, such as plane waves or Bessel functions, are available.
Aeroacoustic models involve equations whose physical characteristics depend on the spatial variables. In general, this PDE system cannot be solved analytically. A natural idea is to resort to basis functions that are approximate solutions of the considered PDE.
In this talk, we consider the simplified model $\Delta u+\kappa^2(x)u=0$. We first present a numerical method to build two families of generalized plane wave bases. The first one is called phase based [1] and takes the following form $\exp(P(x,y))$ where $P(x)=i\kappa_xx+i\kappa_yy+O(x^2+y^2)$ is a complex polynomial functions, with $\kappa_x^2+\kappa_y^2=\kappa^2(x_0)$. The second family is called amplitude based [2]: $Q(x,y)\exp(i\kappa_xx+i\kappa_yy)$ where $Q$ is a complex polynomial such that $Q(0)=1$. Numerical results will illustrate the approximation properties of these functions.
Next we introduce a variational formulation for the boundary value problem, based on a hyperbolic system formulation. It leads to the principle of reciprocity and gives us a formulation similar to the Ultra Weak Variational Formulation [3] when the coefficient $\kappa^2$ is constant.
[1] Lise-Marie Imbert-Gérard. Interpolation properties of generalized plane waves. Numerische Mathematik, 131(4):683-711, December 2015.
[2] Lise-Marie Imbert-Gerard. Amplitude-based Generalized Plane Waves: New Quasi-Trefftz Functions for Scalar Equations in two dimensions. SIAM Journal on Numerical Analysis, 59(3):1663-1686, January 2021.
[3] Olivier Cessenat and Bruno Despres. Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem. SIAM Journal on Numerical Analysis, 35(1):255-299, January 1998.

We introduce a package designed to incorporate Trefftz finite element spaces into NGSolve. This package offers various Trefftz spaces, including harmonic polynomials, plane waves, and caloric polynomials, among others. Moreover, the package introduces unique functionalities, including a quasi-Trefftz space that mimics Trefftz properties for PDEs with smoothcoefficients, space-time Trefftz methods on tent pitched meshes, and a general framework for implicit generation of Trefftz spaces through the embedded Trefftz method.
During our presentation, we will demonstrate how to set up and utilize these features, showing multiple examples. Additionally, we will provide insights into extending the package giving insight into its C++ core.

A new technique (see JJIAM, 2023) which allows the transformation of stabilizations into spaces is described and few applications sketched: 1. How to recast any mixed method for second-order elliptic equations as an HDG method in order to improve its efficiency.
2. How to show that the discretization of the time derivative by the continuous and discontinuous Galerkin methods for ODEs is exactly the same. And immediately obtain superconvergence points of the DG method obtained.
3. How to define the Turbo Post-Processing (TPP) to transform oscillations of the approximation error around zero into new enhanced accuracy approximations for DG methods for convection and for CG methods for diffusion.
As the technique can be applied to any numerical method and any PDE, we argue it should work for Trefftz methods too.

As in [D. Casati and R. Hiptmair, Coupling finite elements and auxiliary sources, Comput. Math. Appl., 77 (2019), pp. 1513-1526], [D. Casati and R. Hiptmair, Coupling FEM with a multiple-subdomain Trefftz method, J. Sci. Comput., 82 (2020), Paper No. 74] and [D. Casati, R. Hiptmair, and J.Smajic, Coupling finite elements and auxiliary sources for electromagnetic wave propagation, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 33 (2020), p. E2752], and, in an abstract framework in [D. Casati, L. Codecasa, R. Hiptmair, and F. Moro, Trefftz co-chain calculus, Z. Angew. Math. Phys., 73 (2022), Paper No. 43], we consider scalar and electromagnetic wave propagation in frequency domain on unbounded domains, partly filled with inhomogeneous media.
We propose a discretization that relies on a Trefftz approximation by multipole auxiliary sources in some parts of the domain and on standard mesh-based primal Lagrangian finite elements in other parts. Several approaches are developed and, based on variational saddle point theory, rigorously analyzed to couple both discretizations across the common interface:
1. Least-squares-based coupling using techniques from PDE-constrained optimization.
2. Coupling through Dirichlet-to-Neumann operators.
3. Three-field variational formulation in the spirit of mortar finite element methods.
We compare these approaches in a series of numerical experiments.

The Ultra Weak Variational Formulation (UWVF) of Maxwell's equations was proposed in Olivier Cessenat's seminal thesis in 1996. It is a Trefftz method based, typically, on the use of plane waves to provide a local approximation of the global solution on a finite element grid. Local solutions are coupled via upwinding across element faces in a variational formulation that enforces inter-element continuity approximately. For real electromagnetic coefficients, the method is a special case of a more general interior penalty Trefftz discontinuous Galerkin method, and this observation underlies an error analysis of the method. I shall give a historical overview of the UWVF for Maxwell's equations including a derivation, summary of the error analysis and comments on the solution of the linear system. I will end with some numerical results concerning the choice of number of plane wave directions per element and a comparison to the finite element method.

The Ultra-Weak Variational Formulation (UWVF) is a Trefftz discontinuous Galerkin method, utilizing superpositions of plane waves for localized solutions on a finite element grid. In the current work, the UWVF is applied to the time-harmonic Maxwell’s equations. Our focus lies on the parallel implementation of UWVF in software named ParMax, and highlighting its efficacy in the simulation of electromagnetic wave problems. Recent enhancements such as the support for various element types, curved elements, and a low memory strategy have a major impact on the software's applicability in industrial problems.
Here, the applicability of the simulation software is examined using various numerical examples. Leveraging large curved elements and the low-memory strategy, we effectively simulated X-band frequency scattering from an aircraft, underscoring the practical utility of ParMax for industrial applications. The talk also includes a discussion of the potential development directions to overcome some current challenges and speed up the computation.

We introduce the concept of an M-decomposition and show how to use it to systematically construct hybridizable discontinuous Galerkin and mixed methods for steady-state diffusion methods with superconvergence properties on unstructured meshes.

Trefftz functions, by definition, satisfy (locally) the underlying differential equation of a given problem, along with the relevant interface boundary conditions. Simple examples include harmonic polynomials for the Laplace equation; plane waves, cylindrical or spherical harmonics for wave problems; exponential functions for the linearized Poisson-Boltzmann equation. At the same time, more complex cases are of great theoretical and practical interest: e.g., waves in disordered structures and Bloch modes in periodic media. The presentation contains two main parts, closely related but somewhat different in spirit. The first part is skewed toward numerical techniques rather than physical phenomena; the second part has the opposite slant.
1. Numerical applications of Trefftz bases.
   - Finite difference Trefftz schemes ("FLAME", the Flexible Local Approximation MEthod), with a variety of tutorial-style examples.
   - Trefftz difference schemes for the Poisson-Boltzmann equation.
   - Trefftz schemes as radiation boundary conditions.
   - Trefftz difference schemes for waves in disordered structures.
   - Trefftz difference schemes for the computation of Bloch bands.
2. Physical applications of Trefftz bases.
   - Non-asymptotic / nonlocal homogenization of periodic structures.
   - Topologically protected boundary modes in electrodynamics.

We present ParaSDG, a parallel-adaptive causal Spacetime Discontinuous Galerkin (cSDG) solver for hyperbolic PDEs -a generalization of Tent Pitching. Adaptive meshing and the numerical solution localize to patches- clusters of spacetime simplex cells such that all patch-boundary facets are space-like. In lieu of traditional domain decomposition, patches act as the unit of parallel execution within an asynchronous, task-based distributed software architecture. This structure supports probabilistic procedures for extremely dynamic data and load balancing. We describe causal adaptive meshing in up to 3d x time and demonstrate how propagating cracks are tracked. Numerical examples are drawn from seismology, fracture mechanics, and electromagnetics. We previously explored a cSDG variant in which single spacetime polytope elements cover entire patch domains. This reduces the number of degrees of freedom per patch (vs. simplex elements) and eliminates the need for Riemann solutions in many problems. Although not yet implemented, we discuss the potential for a cSDG solver using Trefftz basis functions defined over polytope elements. Another connection is made with Trefftz methods for simple 1D x time problems wherein the solution from the inflow facets is directly mapped to the outflow facets using precomputed transfer matrices. We close with a review of schemes that extend cSDG concepts or methods beyond hyperbolic systems. This includes a parabolic-system solver that uses a variant of cSDG spacetime meshing in which stability, rather than the causality constraint limits local time advance. A second scheme uses the cSDG method to find a hyperbolic system’s steady or harmonic-state solution.

Trefftz variational methods, originally introduced by Cessenat and Despres , are discontinuous Galerkin numerical methods whose basis functions are solutions of the underlying partial differential equation that we want to solve numerically. They benefit from a solid theoretical framework that ensures their convergence. These methods can be solved iteratively and define a domain decomposition method. This drastically reduce their memory cost avoiding to resort to a LU decomposition. However, they are polluted by rounding errors that have severely limited their use in 3D.
In this presentation, I will provide a new perspective to these methods in the context of linear hyperbolic problems. This includes a large variety of PDE systems like heterogeneous and anisotropic acoustic, elastic, and Maxwell systems. Recalling the theory of Friedrichs and Rauch, I will explain how to define general boundary conditions for this type of boundary value problems.
Finally, I will explain how these variational methods can be modified to limit the impact of rounding errors. I will present two alternative techniques: a filtering method and a modification of the basis functions known as quasi-Trefftz. The presentation will conclude with concrete illustrations on very large computational scenes.

Part 1 - Code development
Part 2 - Ill conditioning
Part 3 - Applications