A space-time DPG method for the wave equation in multiple dimensions

A space-time discontinuous Petrov–Galerkin (DPG) method for the linear wave equation is presented. This method is based on a weak formulation that uses a broken graph space. The well-posedness of this formulation is established using a previously presented abstract framework. One of the main tasks in the verification of the conditions of this framework is proving a density result. This is done in detail for a simple domain in arbitrary dimensions. The DPG method based on the weak formulation is then studied theoretically and numerically. Error estimates and numerical results are presented for triangular, rectangular, tetrahedral, and hexahedral meshes of the space-time domain. The potential for using the built-in error estimator of the DPG method for an adaptive mesh refinement strategy in two and three dimensions is also presentedThis work was partly supported by AFOSR grant FA9550–17–1–0090. Numerical studies were partially facilitated by the Portland Institute of Sciences (PICS) established under NSF grant DMS–1624776

A space-time discontinuous Petrov- Galerkin method for acousticwaves

We apply the discontinuous Petrov-Galerkin (DPG) method to linear acoustic waves in space and time using the framework of first-order Friedrichs systems. Based on results for operators and semigroups of hyperbolic systems, we show that the ideal DPG method is wellposed. The main task is to avoid the explicit use of traces, which are difficult to define in Hilbert spaces with respect to the graph norm of the space-time differential operator. Then, the practical DPG method is analyzed by constructing a Fortin operator numerically. For our numerical experiments we introduce a simplified DPG method with discontinuous ansatz functions on the faces of the space-time skeleton, where the error is bounded by an equivalent conforming DPG method. Examples for a plane wave configuration confirms the numerical analysis, and the computation of a diffraction pattern illustrates a first step to applications

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

Parallel adaptive discontinuous Galerkin discretizations in space and time for linear elastic and acousticwaves

We introduce a space-time discretization for elastic and acoustic waves using a discontinuous Galerkin approximation in space and a Petrov–Galerkin scheme in time. For the dG method, the upwind flux is evaluated by explicitly solving a Riemann problem. Then we show well-posedness and convergence of the discrete system. Based on goal-oriented dualweighted error estimation an adaptive strategy is introduced. The full space-time linear system is solved with a parallel multilevel preconditioner. Numerical experiments for acoustic and elastic waves underline the efficiency of the overall adaptive solution process

Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century

Es gibt wohl kaum Wissenschaftsgebiete, in denen die wechselseitige Beeinflussung stärker ist als zwischen Mathematik und Physik. Eine wichtige Frage ist dabei die nach der konkreten Ausgestaltung dieser Wechselbeziehungen, etwa an einer Universität, oder die nach prägenden Merkmalen in der Entwicklung dieser Beziehungen in einem historischen Zeitabschnitt. Im Rahmen eines mehrjährigen Akademieprojekts wurden diese Beziehungen an den Universitäten in Leipzig, Halle und Jena für den Zeitraum vom Beginn des 19. bis zur Mitte des 20. Jahrhunderts untersucht und in fünf Bänden dargestellt. Der erste dieser Bände erschien in den Abhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, die nachfolgenden als eigenständige Reihe unter dem Titel “Studien zur Entwicklung von Mathematik und Physik in ihren Wechselwirkungen“. Ein weiterer und abschließender Band dieser Reihe (der vorliegende) beinhaltet die Beiträge einer wissenschaftshistorischen Fachtagung im Jahr 2010, die das Thema in einem internationalen Kontext einbettet. Der vorliegende Band enthält die Beiträge der Tagung “Mathematics meets physics. A contribution to their interaction in the 19th and the first half of the 20th century”, die vom 22. bis 25. März 2010 in Leipzig stattfand. Die Konferenzbeiträge bestätigen die große Variabilität in der Gestaltung der Wechselbeziehungen zwischen Mathematik und Physik. In ihnen werden u.a. verschiedene Entwicklungsprozesse im 19. und 20. Jahrhundert (zur elektromagnetischen Feldtheorie, zur Quantenmechanik, zur Quantenfeldtheorie, zur Relativitätstheorie) aus unterschiedlichen Perspektiven analysiert. Weitere Beiträge stellen allgemeinere Fragestellungen der Entwicklung der Wechselbeziehungen in den Mittelpunkt und tragen zur Frage einer möglichen Unterscheidung unterschiedlicher Entwicklungsstufen im den Wechselverhältnis von Mathematik und Physik bei. Insgesamt ist einzuschätzen: Zum einen dokumentieren die in den Beiträgen vorgelegten Ergebnisse den Wert und die Notwendigkeit von Detailuntersuchungen, um die Entwicklung der Wechselbeziehungen zwischen Mathematik und Physik in ihrer Vielfalt und mit der nötigen Präzision zu erfassen, zum anderen lassen sie in ihrer Gesamtheit noch zu beantwortende Forschungsfragen erkennen.:Vorwort Karl-Heinz Schlote, Martina Schneider: Introduction Jesper Lützen: Examples and Reflections on the Interplay between Mathematics and Physics in the 19th and 20th Century Juraj Šebesta: Mathematics as one of the basic Pillars of physical Theory: a historical and epistemological Survey Karl-Heinz Schlote, Martina Schneider: The Interrelation between Mathematics and Physics at the Universities Jena, Halle-Wittenberg and Leipzig – a Comparison Karin Reich: Der erste Professor für Theoretische Physik an der Universität Hamburg: Wilhelm Lenz Jim Ritter: Geometry as Physics: Oswald Veblen and the Princeton School Erhard Scholz: Mathematische Physik bei Hermann Weyl – zwischen „Hegelscher Physik“ und „symbolischer Konstruktion der Wirklichkeit“ Scott Walter: Henri Poincaré, theoretical Physics, and Relativity Theory in Paris Reinhard Siegmund-Schultze: Indeterminismus vor der Quantenmechanik: Richard von Mises’ wahrscheinlichkeitstheoretischer Purismus in der Theorie physikalischer Prozesse Christoph Lehner: Mathematical Foundations and physical Visions: Pascual Jordan and the Field Theory Program Jan Lacki: From Matrices to Hilbert Spaces: The Interplay of Physics and Mathematics in the Rise of Quantum Mechanics Helge Kragh: Mathematics, Relativity, and Quantum Wave Equations Klaus-Heinrich Peters: Mathematische und phänomenologische Strenge: Distributionen in der Quantenmechanik und -feldtheorie Arianna Borrelli: Angular Momentum between Physics and Mathematics Friedrich Steinle: Die Entstehung der Feldtheorie: ein ungewöhnlicher Fall der Wechselwirkung von Physik und Mathematik? Vortragsprogramm Liste der Autoren Personenverzeichni

Space-Time Methods for Acoustic Waves with Applications to Full Waveform Inversion

Classically, wave equations are considered as evolution equations where the derivative with respect to time is treated in a stronger way than the spatial differential operators. This results in an ordinary differential equation (ODE) with values in a function space, e.g. in a Hilbert space, with respect to the spatial variable. For instance, acoustic waves in a spatial domain $\Omega \subset \mathbb{R}^d$ for a given right-hand side $\mathbf b$ can be considered in terms of the following ODE \begin{equation*} \partial_t \mathbf y = A\mathbf y + \mathbf b\quad \text{ in }[0,T]\,,\quad \mathbf y(0) = \mathbf 0\,, \qquad A = \begin{pmatrix} 0 & \operatorname{div} \\ \nabla & 0 \end{pmatrix}, \end{equation*} where the solution $\mathbf y = (p, \mathbf v)$ is an element of the space $\mathrm C^0\big(0,T; \mathcal D(A)\big) \cap \mathrm C^1\big(0,T; \mathrm L_2(\Omega)\big)$ with $\mathcal D(A) \subset \mathrm H^1(\Omega) \times H(\operatorname{div}, \Omega)$. In order to analyze this ODE, space and time are treated separately and hence tools for partial differential equations are used in space and tools for ODEs are used in time. Typically, this separation carries over to the analysis of numerical schemes to approximate solutions of the equation. By contrast, in this work, we consider the space-time operator \begin{equation*} L (p,\mathbf v) = \begin{pmatrix} \partial_t p + \operatorname{div} \mathbf v \\ \partial_t \mathbf v + \nabla p \end{pmatrix}\,, \end{equation*} in $Q = (0,T) \times \Omega$ as a whole treating time and space dependence simultaneously in a variational manner. Using this approach, we constructed a space-time Hilbert space setting that allows for irregular solutions, e.g. with space-time discontinuities. Within this variational framework, we construct and analyzed two classes of non-conforming discretization schemes for acoustic waves, a Discontinuous Petrov-Galerkin method and a scheme of Least-Squares type. For both methods, we provide a convergence analysis exploiting tools from classical Finite Element theory for space and also time dependence. The theoretical predictions are complemented by extensive numerical experiments showing that high convergence rates are attained in practice. While considering the problem of Full Waveform Inversion (FWI), we focus on the derivation of Newton-type algorithms to tackle this inverse problem numerically. Here, we make extensive use of the space-time $\mathrm L_2(Q)$ adjoint $L^*$ that is easily accessible within our variational space-time framework. We implement a regularized inexact Newton method, CG-REGINN, and provide a numerical example for a benchmark problem