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1. Stability analysis for electromagnetic waveguides. Part 2: non-homogeneous waveguides

   https://doi.org/10.1007/s10444-024-10130-x  

   Demkowicz, Leszek; Melenk, Jens M; Badger, Jacob; Henneking, Stefan (April 2024, Advances in Computational Mathematics)

   This paper is a continuation of Melenk et al., "Stability analysis for electromagnetic waveguides. Part 1: acoustic and homogeneous electromagnetic waveguides" (2023), extending the stability results for homogeneous electromagnetic (EM) waveguides to the non-homogeneous case. The analysis is done using perturbation techniques for self-adjoint operators eigenproblems. We show that the non-homogeneous EM waveguide problem is well-posed with the stability constant scaling linearly with waveguide length L. The results provide a basis for proving convergence of a Discontinuous Petrov-Galerkin (DPG) discretization based on a full envelope ansatz, and the ultraweak variational formulation for the resulting modified system of Maxwell equations, see Part 1. 

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2. The discontinuous Petrov–Galerkin method

   https://doi.org/10.1017/S0962492924000102  

   Demkowicz, Leszek; Gopalakrishnan, Jay (July 2025, Acta Numerica)

   The discontinuous Petrov–Galerkin (DPG) method is a Petrov–Galerkin finite element method with test functions designed for obtaining stability. These test functions are computable locally, element by element, and are motivated by optimal test functions which attain the supremum in an inf-sup condition. A profound consequence of the use of nearly optimal test functions is that the DPG method can inherit the stability of the (undiscretized) variational formulation, be it coercive or not. This paper combines a presentation of the fundamentals of the DPG ideas with a review of the ongoing research on theory and applications of the DPG methodology. The scope of the presented theory is restricted to linear problems on Hilbert spaces, but pointers to extensions are provided. Multiple viewpoints to the basic theory are provided. They show that the DPG method is equivalent to a method which minimizes a residual in a dual norm, as well as to a mixed method where one solution component is an approximate error representation function. Being a residual minimization method, the DPG method yields Hermitian positive definite stiffness matrix systems even for non-self-adjoint boundary value problems. Having a built-in error representation, the method has the out-of-the-box feature that it can immediately be used in automatic adaptive algorithms. Contrary to standard Galerkin methods, which are uninformed about test and trial norms, the DPG method must be equipped with a concrete test norm which enters the computations. Of particular interest are variational formulations in which one can tailor the norm to obtain robust stability. Key techniques to rigorously prove convergence of DPG schemes, including construction of Fortin operators, which in the DPG case can be done element by element, are discussed in detail. Pointers to open frontiers are presented. 

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3. A vectorial envelope Maxwell formulation for electromagnetic waveguides with application to nonlinear fiber optics

   https://doi.org/10.1016/j.camwa.2025.05.026  

   Henneking, Stefan; Grosek, Jacob; Demkowicz, Leszek (September 2025, Computers & Mathematics with Applications)

   This article presents an ultraweak discontinuous Petrov-Galerkin (DPG) formulation of the time-harmonic Maxwell equations for the vectorial envelope of the electromagnetic field in a weakly-guiding multi-mode fiber waveguide. This formulation is derived using an envelope ansatz for the vector-valued electric and magnetic field components, factoring out an oscillatory term of exp(-ikz) with a user-defined wavenumber k, where z is the longitudinal fiber axis and field propagation direction. The resulting formulation is a modified system of the time-harmonic Maxwell equations for the vectorial envelope of the propagating field. This envelope is less oscillatory in the z-direction than the original field, so that it can be more efficiently discretized and computed, enabling solutions to the vectorial DPG Maxwell system in fibers that are 1000x longer than previously possible. Different approaches for incorporating a perfectly matched layer for absorbing the outgoing wave modes at the fiber end are derived and compared numerically. The resulting formulation is used to solve a 3D Maxwell model of an ytterbium-doped active gain fiber amplifier, coupled with the heat equation for including thermal effects. The nonlinear model is then used to simulate thermally-induced transverse mode instability (TMI). The numerical experiments demonstrate that it is computationally feasible to perform simulations and analysis of real-length optical fiber laser amplifiers using discretizations of the full vectorial time-harmonic Maxwell equations. The approach promises a new high-fidelity methodology for analyzing TMI in high-power fiber laser systems and is extendable to including other nonlinearities. 

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4. Equivalence between the DPG method and the exponential integrators for linear parabolic problems

   https://doi.org/10.1016/j.jcp.2020.110016  

   Munoz-Matute, J.; Pardo, D.; Demkowicz, L. (July 2021, Journal of computational physics)

   The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well establishednumerical methods for solving Partial Differential Equations (PDEs) and stiff systems of Ordinary Differential Equations (ODEs), respectively. In this work, we apply the DPG method in the time variable for linear parabolic problems and we calculate the optimal test functions analytically. We show that the DPG method in time is equivalent to exponential integrators for the trace variables, which are decoupled from the interior variables. In addition, the DPG optimal test functions allow us to compute the approximated solutions in the time element interiors. This DPG method in time allows to construct a posteriori error estimations in order to perform adaptivity. We generalize this novel DPG-based time-marching scheme to general first order linear systems of ODEs. We show the performance of the proposed method for 1D and 2D +time linear parabolic PDEs after discretizing in space by the finite element method. 

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5. Scalable DPG multigrid solver with applications in high-frequency wave propagation

   https://doi.org/10.26153/tsw/55685  

   Badger, Jacob (January 2024, The University of Texas at Austin)

   Wave propagation is fundamental to applications including natural resource exploration, nuclear fusion research, and military defense, among others. However, developing accurate and efficient numerical algorithms for solving time-harmonic wave propagation problems is notoriously difficult. One difficulty is that classical discretization techniques (e.g., Galerkin finite elements, finite difference, etc.) yield indefinite discrete systems that preclude the use of many scalable solution algorithms. Significant progress has been made to develop specialized preconditioners for high-frequency wave propagation problems but robust and scalable solvers for general problems, including non-homogenous media and complex geometries, remain elusive. An alternative approach is to use minimum residual discretization methods—that yield Hermitian positive-definite discrete systems—and may be amenable to more standard preconditioners. Indeed, popularization of the first-order system least-squares methodology (FOSLS) was driven by the applicability of geometric and algebraic multigrid to otherwise indefinite problems. However, for wave propagation problems, FOSLS is known to be highly dissipative and is thus less competitive in the high-frequency regime. The discontinuous Petrov–Galerkin (DPG) method of Demkowicz and Gopalakrishnan is a minimum residual finite element method with several additional attractive properties: mesh-independent stability, a built-in error indicator, and applicability to a number of variational formulations. In the context of high-frequency wave propagation, the ultraweak DPG formulation has been observed to produce pollution error roughly commensurate to Galerkin discretizations. DPG discretizations may thus deliver accuracy typical of classical discretization techniques, but result in Hermitian positive-definite discrete systems that are often more amenable to preconditioning. A multigrid preconditioner for DPG systems, developed in the dissertation work of S. Petrides, was shown to scale efficiently in a shared-memory implementation. The primary objective of this dissertation is development of an efficient, distributed implementation of the DPG multigrid solver (DPG-MG). The distributed DPG-MG solver developed in this work will be demonstrated to be massively scalable, enabling solution of three-dimensional problems with O(10¹²) degrees of freedom on up to 460 000 CPU cores, an unprecedented scale for high-frequency wave propagation. The scalability of the DPG-MG solver will be further combined with hp-adaptivity to enable efficient solution of challenging real-world high-frequency wave propagation problems including optical fiber modeling, simulation of RF heating in tokamak devices, and seismic simulation. These applications include complex three-dimensional geometries, heterogeneous and anisotropic media, and localized features; demonstrating the robustness and versatility of the solver and tools developed in this dissertation. 

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