arXiv subjects

Leszek Demkowicz

Publications and source records attributed to Leszek Demkowicz.

At least 19 recordsLinked to original sources

arXiv2026

Bent optical waveguide finite element analysis with a 3D envelope Maxwell model

With the goal of accurately extracting the optical field losses in a three-dimensional (3D), circularly coiled waveguide (e.g., bent optical fiber), this effort presents the numerical methodologies that are implemented for an envelope Maxwell model that propagates electromagnetic fields as an entirely boundary value problem. Our unique modeling approach includes an ultraweak variational formulation of the envelope Maxwell model in the curved geometry of the bending, which is discretized by the discontinuous Petrov-Galerkin (DPG) method, which permits residual-driven mesh and polynomial-order adaptivity. This also, then, requires a unique approach for constructing perfectly matched layers (PMLs) as absorbing boundary conditions in both the direction of optical field propagation and in the tangential directions, where unguided energy escapes the waveguide. Our coiled waveguide modeling technology extracts the mode confinement losses from the propagation of the coherent optical field through the bent waveguide. We verify our simulations against the semi-analytical results from the analogous bent slab waveguide problem, and we successfully demonstrate stable convergence to loss values for the 3D coiled optical fiber problem, which has never been done previously for our specific modeling approach.

physics.optics↗

arXiv2026

Length-explicit stability analysis of Helmholtz problems in leaky circular waveguides

Motivated by the study and simulation of long, coiled optical fibers we consider in this article a simplified model that is prevalent in the engineering community. Mathematically, the problem is specified as follows: Time-harmonic wave propagation is modeled by the Helmholtz equation; the waveguide is a bounded circular section with a transparent boundary condition on one end; the dissipation of energy is modeled by an impedance boundary condition on the outer hull of the waveguide. We show a stability estimate that is explicit in terms of the angular length of the waveguide. The analysis is based on a separation of variables ansatz and the study of the related (nonselfadjoint) modal eigenvalue problem. The key property there is to show that the modes form a Riesz basis in both $L^2$ and $H^1$ spaces. To this end we apply perturbation theory for selfadjoint operators and the concept of local subordination of perturbations [B. Mityagin and P. Siegl, JAM 139 (2019)]. Since the possibility of nontrivial Jordan chains cannot be ruled out, our whole methodology is conducted accordingly. In addition, in contrast to previous works, we include a bounded but heterogeneous part of the waveguide into our considered setting.

math.AP↗

arXiv2025

Bessel Functions and Analysis of Circular Waveguides

The paper is devoted to the study of circularly coiled optical slab waveguides, which is also applicable to acoustical waveguides. We use a change of variables and the classical Frobenius method to compute Bessel functions of complex order and complex argument, and combine it with a perfectly matched layer technique to solve the relevant Bessel eigenvalue problem and deliver accurate loss factors for eigensolutions to the three-layer optical slab waveguide problem. The solutions provide a benchmark for verifying model implementations of this problem and allow for a numerical verification of the Glazman criterion that provides a foundation for the well-posedness and stability analysis of homogeneous circular waveguides with impedance boundary conditions.

physics.comp-ph↗

arXiv2024

A Space-Time Discontinuous Petrov-Galerkin Finite Element Formulation for a Modified Schr\"odinger Equation for Laser Pulse Propagation in Waveguides

In this article, we propose a modified nonlinear Schr\"odinger equation for modeling pulse propagation in optical waveguides. The proposed model bifurcates into a system of elliptic and hyperbolic equations depending on waveguide parameters. The proposed model leads to a stable first-order system of equations, distinguishing itself from the canonical nonlinear Schr\"odinger equation. We have employed the space-time discontinuous Petrov-Galerkin finite element method to discretize the first-order system of equations. We present a stability analysis for both the elliptic and hyperbolic systems of equations and demonstrate the stability of the proposed model through several numerical examples on space-time meshes.

math.NA↗

arXiv2024

A Vectorial Envelope Maxwell Formulation for Electromagnetic Waveguides with Application to Nonlinear Fiber Optics

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 $exp(-i \mathsf{k}z)$ with a user-defined wavenumber $\mathsf{k}$, 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 solution of the vectorial DPG Maxwell system for $1000\times$ longer fibers 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.

math.NA↗

arXiv2024

Multistage DPG time-marching scheme for nonlinear problems

In this article, we employ the construction of the time-marching Discontinuous Petrov-Galerkin (DPG) scheme we developed for linear problems to derive high-order multistage DPG methods for non-linear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach spaces, including a class of nonlinear partial differential equations. We present three nested multistage methods: the hybrid Euler method and the two- and three-stage DPG methods. We employ a linearization of the problem as in exponential Rosenbrock methods, so we need to compute exponential actions of the Jacobian that change from time steps. The key point of our construction is that one of the stages can be post-processed from another without an extra exponential step. Therefore, the class of methods we introduce is computationally cheaper than the classical exponential Rosenbrock methods. We provide a full convergence proof to show that the methods are second, third, and fourth-order accurate, respectively. We test the convergence in time of our methods on a 2D + time semi-linear partial differential equation after a semidiscretization in space.

math.NA↗

arXiv2023

Scalable DPG Multigrid Solver for Helmholtz Problems: A Study on Convergence

This paper presents a scalable multigrid preconditioner targeting large-scale systems arising from discontinuous Petrov-Galerkin (DPG) discretizations of high-frequency wave operators. This work is built on previously developed multigrid preconditioning techniques of Petrides and Demkowicz (Comput. Math. Appl. 87 (2021) pp. 12-26) and extends the convergence results from $\mathcal{O}(10^7)$ degrees of freedom (DOFs) to $\mathcal{O}(10^9)$ DOFs using a new scalable parallel MPI/OpenMP implementation. Novel contributions of this paper include an alternative definition of coarse-grid systems based on restriction of fine-grid operators, yielding superior convergence results. In the uniform refinement setting, a detailed convergence study is provided, demonstrating h and p robust convergence and linear dependence with respect to the wave frequency. The paper concludes with numerical results on hp-adaptive simulations including a large-scale seismic modeling benchmark problem with high material contrast.

math.NA↗

arXiv2023

hp3D User Manual

User Manual for the hp3D Finite Element Software, available on GitHub at https://github.com/Oden-EAG/hp3d

math.NA↗

arXiv2023

An Anisotropic $hp$-Adaptation Framework for Ultraweak Discontinuous Petrov-Galerkin Formulations

In this article, we present a three-dimensional anisotropic $hp$-mesh refinement strategy for ultraweak discontinuous Petrov--Galerkin (DPG) formulations with optimal test functions. The refinement strategy utilizes the built-in residual-based error estimator accompanying the DPG discretization. The refinement strategy is a two-step process: (a) use the built-in error estimator to mark and isotropically $hp$-refine elements of the (coarse) mesh to generate a finer mesh; (b) use the reference solution on the finer mesh to compute optimal $h$- and $p$-refinements of the selected elements in the coarse mesh. The process is repeated with coarse and fine mesh being generated in every adaptation cycle, until a prescribed error tolerance is achieved. We demonstrate the performance of the proposed refinement strategy using several numerical examples on hexahedral meshes.

cs.CE↗

arXiv2023

Stability Analysis for Electromagnetic Waveguides. Part 1: Acoustic and Homogeneous Electromagnetic Waveguides

In a time-harmonic setting, we show for heterogeneous acoustic and homogeneous electromagnetic wavesguides stability estimates with the stability constant depending linearly on the length $L$ of the waveguide. These stability estimates are used for the analysis of the (ideal) ultraweak (UW) variant of the Discontinuous Petrov Galerkin (DPG) method. For this UW DPG, we show that the stability deterioration with $L$ can be countered by suitably scaling the test norm of the method. We present the ``full envelope approximation'', a UW DPG method based on non-polynomial ansatz functions that allows for treating long waveguides.

math.NA↗

arXiv2022

Deep neural networks for smooth approximation of physics with higher order and continuity B-spline base functions

This paper deals with the following important research question. Traditionally, the neural network employs non-linear activation functions concatenated with linear operators to approximate a given physical phenomenon. They "fill the space" with the concatenations of the activation functions and linear operators and adjust their coefficients to approximate the physical phenomena. We claim that it is better to "fill the space" with linear combinations of smooth higher-order B-splines base functions as employed by isogeometric analysis and utilize the neural networks to adjust the coefficients of linear combinations. In other words, the possibilities of using neural networks for approximating the B-spline base functions' coefficients and by approximating the solution directly are evaluated. Solving differential equations with neural networks has been proposed by Maziar Raissi et al. in 2017 by introducing Physics-informed Neural Networks (PINN), which naturally encode underlying physical laws as prior information. Approximation of coefficients using a function as an input leverages the well-known capability of neural networks being universal function approximators. In essence, in the PINN approach the network approximates the value of the given field at a given point. We present an alternative approach, where the physcial quantity is approximated as a linear combination of smooth B-spline basis functions, and the neural network approximates the coefficients of B-splines. This research compares results from the DNN approximating the coefficients of the linear combination of B-spline basis functions, with the DNN approximating the solution directly. We show that our approach is cheaper and more accurate when approximating smooth physical fields.

math.NA↗

arXiv2021

Stabilized finite element methods for a fully-implicit logarithmic reformulation of the Oldroyd-B constitutive law

Logarithmic conformation reformulations for viscoelastic constitutive laws have alleviated the high Weissenberg number problem, and the exploration of highly elastic flows became possible. However, stabilized formulations for logarithmic conformation reformulations in the context of finite element methods have not yet been widely studied. We present stabilized formulations for the logarithmic reformulation of the Oldroyd-B model by Saramito (2014) based on the Variational Multiscale framework and Galerkin/Least-Squares method. The reformulation allows the use of Newton's method due to its fully-implicit nature for solving the steady-state problem directly. The proposed stabilization methods cure instabilities of convection and compatibility while preserving a three-field problem. The spatial accuracy of the formulations is assessed with the four-roll periodic box, revealing comparable accuracy between the methods. The formulations are validated with benchmark flows past a cylinder and through a 4:1 contraction. We found an excellent agreement to benchmark results in the literature. The algebraic sub-grid scale method is highly robust, indicated by the high limiting Weissenberg numbers in the benchmark flows.

cs.CE↗

arXiv2021

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

We present a Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al.~(2012), we embed the original nonlocal convection-dominated diffusion problem into a larger mixed problem so as to choose an enriched test space as a stabilization of the numerical algorithm. In the numerical experiments, we use an approximate optimal test norm which can be efficiently implemented in 1d, and study its performance against the energy norm on the test space. We conduct convergence studies for the nonlocal problem using uniform $h$- and $p$-refinements, and adaptive $h$-refinements on both smooth manufactured solutions and solutions with sharp gradient in a transition layer. In addition, we confirm that the PG method is asymptotically compatible.

math.NA↗

arXiv2020

An adaptive multigrid solver for DPG methods with applications in linear acoustics and electromagnetics

We propose an adaptive multigrid preconditioning technology for solving linear systems arising from Discontinuous Petrov-Galerkin (DPG) discretizations. Unlike standard multigrid techniques, this preconditioner involves only trace spaces defined on the mesh skeleton, and it is suitable for adaptive hp-meshes. The key point of the construction is the integration of the iterative solver with a fully automatic and reliable mesh refinement process provided by the DPG technology. The efficacy of the solution technique is showcased with numerous examples of linear acoustics and electromagnetic simulations, including simulations in the high-frequency regime, problems which otherwise would be intractable. Finally, we analyze the one-level preconditioner (smoother) for uniform meshes and we demonstrate that theoretical estimates of the condition number of the preconditioned linear system can be derived based on well established theory for self-adjoint positive definite operators.

math.NA↗

arXiv2020

A numerical study of the pollution error and DPG adaptivity for long waveguide simulations

High-frequency wave propagation has many important applications in acoustics, elastodynamics, and electromagnetics. Unfortunately, the finite element discretization for these problems suffers from significant numerical pollution errors that increase with the wavenumber. It is critical to control these errors to obtain a stable and accurate method. We study the effect of pollution for very long waveguide problems in the context of robust discontinuous Petrov-Galerkin (DPG) finite element discretizations. Our numerical experiments show that the pollution primarily has a diffusive effect causing energy loss in the DPG method while phase errors appear less significant. We report results for 3D vectorial time-harmonic Maxwell problems in waveguides with more than 8000 wavelengths. Our results corroborate previous analysis for the Galerkin discretization of the Helmholtz operator by Melenk and Sauter (2011). Additionally, we discuss adaptive refinement strategies for multi-mode fiber waveguides where the propagating transverse modes must be resolved sufficiently. Our study shows the applicability of the DPG error indicator to this class of problems. Finally, we illustrate the importance of load balancing in these simulations for distributed-memory parallel computing.

math.NA↗

arXiv2020

Model and computational advancements to full vectorial Maxwell model for studying fiber amplifiers

We present both modeling and computational advancements to a unique three-dimensional discontinuous Petrov-Galerkin finite element model for the simulation of laser amplification in a fiber amplifier. Our model is based on the time-harmonic Maxwell equations, and it incorporates both amplification via an active dopant and thermal effects via coupling with the heat equation. As a full vectorial finite element simulation, this model distinguishes itself from other fiber amplifier models that are typically posed as an initial value problem and make significantly more approximations. Our model supports co-, counter-, and bi-directional pumping configurations, as well as inhomogeneous and anisotropic material properties. The longer-term goal of this modeling effort is to study nonlinear phenomena that prohibit achieving unprecedented power levels in fiber amplifiers, along with validating typical approximations used in lower-fidelity models. The high-fidelity simulation comes at the cost of a high-order finite element discretization with many degrees of freedom per wavelength. This is necessary to counter the effect of numerical pollution due to the high-frequency nature of the wave simulation. To make the computation more feasible, we have developed a novel longitudinal model rescaling, using artificial material parameters with the goal of preserving certain quantities of interest. Our numerical tests demonstrate the applicability and utility of this scaled model in the simulation of an ytterbium-doped, step-index fiber amplifier that experiences laser amplification and heating. We present numerical results for the nonlinear coupled Maxwell/heat model with up to 240 wavelengths.

math.NA↗

arXiv2018

The DPG-star method

This article introduces the DPG-star (from now on, denoted DPG$^*$) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG$^*$ methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG$^*$ and DPG methods can be seen as generalizations of $\mathcal{L}\mathcal{L}^\ast$ and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG$^*$ method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG$^*$ and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.

math.NA↗

arXiv2018

Goal-oriented adaptive mesh refinement for non-symmetric functional settings

In this article, a new unified duality theory is developed for Petrov-Galerkin finite element methods. This novel theory is then used to motivate goal-oriented adaptive mesh refinement strategies for use with discontinuous Petrov-Galerkin (DPG) methods. The focus of this article is mainly on broken ultraweak variational formulations of stationary boundary value problems, however, many of the ideas presented within are general enough that they be extended to any such well-posed variational formulation. The proposed goal-oriented adaptive mesh refinement procedures require the construction of refinement indicators for both a primal problem and a dual problem. In the DPG context, the primal problem is simply the system of linear equations coming from a standard DPG method and the dual problem is a similar system of equations, coming from a new method which is dual to DPG. This new method has the same coefficient matrix as the associated DPG method but has a different load. We refer to this new finite element method as a DPG* method. A thorough analysis of DPG* methods, as stand-alone finite element methods, is not given here but will be provided in subsequent articles. For DPG methods, the current theory of a posteriori error estimation is reviewed and the reliability estimate in [13, Theorem 2.1] is improved on. For DPG* methods, three different classes of refinement indicators are derived and several contributions are made towards rigorous a posteriori error estimation. At the closure of the article, results of numerical experiments with Poisson's boundary value problem in a three-dimensional domain are provided. These results clearly demonstrate the utility of the goal-oriented adaptive mesh refinement strategies for quantities of interest with either interior or boundary terms.

math.NA↗