A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D

Abstract

a b s t r a c tThe phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for onedimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov-Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L 2 -norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.

The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

“…Based on the numerical results of others [29,30,37], we observe that the new sd-scheme is better than the finite difference methods and standard finite element by comparing the numerical solution with the exact solution. The agreement of the error estimates between theoretical analysis and numerical results shows that our method is efficient.…”

“…Recently, various numerical methods such as finite difference methods [22], the finite element (fe) method [23], continuous Galerkin (cg) method [20], discontinuous Galerkin (dg) method [31], and discontinuous Petrov-Galerkin (dpg) with optimal test functions [37] have been widely used to solve this problem. However, to the best of our knowledge, only some did numerical work concerning these coupled equations using the finite difference CrankNicolson (cn) scheme and the standard finite element method, while the E56 intensive analysis of the precision of this method is very limited.…”

“…In addition to passing-through collision is an E53 important phenomena such that vector solitons also bounce off each other or trap each other. Solutions of this system have been studied intensively in recent years in numerical aspects and analytical [19,20,21,22,23,24,31,32,33,34,35,36,37]. Numerical examples were investigated by many authors in the one dimensional case [21,21,22,23,24,34].…”

Dimensionally adaptive hp-finite element simulation and inversion of 2D magnetotelluric measurements

“…These methods have been extensively employed in the last years in applied mathematics to solve a variety of engineering problems. For example, the isogeometric analysis (IGA) [8,9] has recently experimented a huge explosion and it has been widely applied to the engineering industry, as well as the more recent Discontinuous Petrov-Galerkin (DPG) method initially proposed by Demkowicz and Gopalakrishnan [10,11], or the self-adaptive hp-Finite Element Method (FEM) [12,13] (where h stands for the element size and p for the polynomial order of approximation associated to each element). The latter one has been recently employed, for instance, to model the bone conduction of sound in the human head [14], or to simulate bend, step, and magic-T electromagnetic waveguide structures [15].…”

A toolbox for a class of discontinuous Petrov-Galerkin methods using trilinos.

“…Recently, L. Demkowicz and J. Gopalakrishnan have proposed a new class of discontinuous Petrov-Galerkin (DPG) methods [4,5,6,9,3], which compute test functions that are adapted to the problem of interest to produce stable discretization schemes. An important choice that must be made in the application of the method is the definition of the inner product on the test space.…”