HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems

Towards exponentially-convergent simulations of extreme-mass-ratio inspirals: A time-domain solver for the scalar Teukolsky equation with singular source terms

Gravitational wave signals from extreme mass ratio inspirals are a key target for space-based gravitational wave detectors. These systems are typically modeled as a distributionally-forced Teukolsky equation, where the smaller black hole is treated as a Dirac delta distribution. Time-domain solvers often use regularization approaches that approximate the Dirac distribution that often introduce small length scales and are a source of systematic error, especially near the smaller black hole. We describe a multi-domain discontinuous Galerkin method for solving the distributionally-forced Teukolsky equation that describes scalar fields evolving on a Kerr spacetime. To handle the Dirac delta, we expand the solution in spherical harmonics and recast the sourced Teukolsky equation as a first-order, one-dimensional symmetric hyperbolic system. This allows us to derive the method's numerical flux to correctly account for the Dirac delta. As a result, our method achieves global spectral accuracy even at the source's location. To connect the near field to future null infinity, we use the hyperboloidal layer method, allowing us to supply outer boundary conditions and providing direct access to the far-field waveform. We document several numerical experiments where we test our method, including convergence tests against exact solutions, energy luminosities for circular orbits, the scheme's superconvergence properties at future null infinity, and the late-time tail behavior of the scalar field. We also compare two systems that arise from different choices of the first-order reduction variables, finding that certain choices are numerically problematic in practice. The methods developed here may be beneficial when computing gravitational self-force effects, where the regularization procedure has been developed for the spherical harmonic modes and high accuracy is needed at the Dirac delta's location.Comment: 20 pages, 7 figures and 2 table

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

Uniform L∞L^\infty-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation

In this paper, we consider an energy-conserving continuous Galerkin discretization of the Gross-Pitaevskii equation with a magnetic trapping potential and a stirring potential for angular momentum rotation. The discretization is based on finite elements in space and time and allows for arbitrary polynomial orders. It was first analyzed in [O. Karakashian, C. Makridakis; SIAM J. Numer. Anal. 36(6):1779-1807, 1999] in the absence of potential terms and corresponding a priori error estimates were derived in 2D. In this work we revisit the approach in the generalized setting of the Gross-Pitaevskii equation with rotation and we prove uniform $L^\infty$-bounds for the corresponding numerical approximations in 2D and 3D without coupling conditions between the spatial mesh size and the time step size. With this result at hand, we are in particular able to extend the previous error estimates to the 3D setting while avoiding artificial CFL conditions

The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications

International audienceHybrid High-Order (HHO) methods are new generation numerical methods for models based on Partial Differential Equations with features that set them apart from traditional ones. These include: the support of polytopal meshes including non star-shaped elements and hanging nodes; the possibility to have arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to compact stencil and static condensation. This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics. The first part of the monograph lays the foundation of the method considering linear scalar second-order models, including scalar diffusion, possibly heterogeneous and anisotropic, and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity and incompressible fluid flows

Solving forward and inverse Helmholtz equations via controllability methods

Waves are useful for probing an unknown medium by illuminating it with a source. To infer the characteristics of the medium from (boundary) measurements, for instance, one typically formulates inverse scattering problems in frequency domain as a PDE-constrained optimization problem. Finding the medium, where the simulated wave field matches the measured (real) wave field, the inverse problem requires the repeated solutions of forward (Helmholtz) problems. Typically, standard numerical methods, e.g. direct solvers or iterative methods, are used to solve the forward problem. However, large-scaled (or high-frequent) scattering problems are known being competitive in computation and storage for standard methods. Moreover, since the optimization problem is severely ill-posed and has a large number of local minima, the inverse problem requires additional regularization akin to minimizing the total variation. Finding a suitable regularization for the inverse problem is critical to tackle the ill-posedness and to reduce the computational cost and storage requirement. In my thesis, we first apply standard methods to forward problems. Then, we consider the controllability method (CM) for solving the forward problem: it instead reformulates the problem in the time domain and seeks the time-harmonic solution of the corresponding wave equation. By iteratively reducing the mismatch between the solution at initial time and after one period with the conjugate gradient (CG) method, the CMCG method greatly speeds up the convergence to the time-harmonic asymptotic limit. Moreover, each conjugate gradient iteration solely relies on standard numerical algorithms, which are inherently parallel and robust against higher frequencies. Based on the original CM, introduced in 1994 by Bristeau et al., for sound-soft scattering problems, we extend the CMCG method to general boundary-value problems governed by the Helmholtz equation. Numerical results not only show the usefulness, robustness, and efficiency of the CMCG method for solving the forward problem, but also demonstrate remarkably accurate solutions. Second, we formulate the PDE-constrained optimization problem governed by the inverse scattering problem to reconstruct the unknown medium. Instead of a grid-based discrete representation combined with standard Tikhonov-type regularization, the unknown medium is projected to a small finite-dimensional subspace, which is iteratively adapted using dynamic thresholding. The adaptive (spectral) space is governed by solving several Poisson-type eigenvalue problems. To tackle the ill-posedness that the Newton-type optimization method converges to a false local minimum, we combine the adaptive spectral inversion (ASI) method with the frequency stepping strategy. Numerical examples illustrate the usefulness of the ASI approach, which not only efficiently and remarkably reduces the dimension of the solution space, but also yields an accurate and robust method