By Aurora Marica, Enrique Zuazua
This paintings describes the propagation homes of the so-called symmetric inside penalty discontinuous Galerkin (SIPG) approximations of the 1-d wave equation. this can be performed through linear approximations on uniform meshes. First, a cautious Fourier research is built, highlighting the coexistence of 2 Fourier spectral branches or spectral diagrams (physical and spurious) on the topic of the 2 parts of the numerical answer (averages and jumps). effective filtering mechanisms also are constructed via suggestions formerly proved to be applicable for classical schemes like finite changes or P1-classical finite components. specifically, the paintings provides an evidence that the uniform observability estate is recovered uniformly by way of contemplating preliminary facts with null jumps and averages given by way of a bi-grid filtering set of rules. ultimately, the publication explains how those effects could be prolonged to different extra subtle conforming and non-conforming finite point tools, particularly to quadratic finite parts, neighborhood discontinuous Galerkin equipment and a model of the SIPG approach including penalization at the common derivatives of the numerical answer on the grid issues. This paintings is the 1st ebook to comprise a rigorous research of the discontinuous Galerkin tools for wave keep watch over difficulties. it is going to be of curiosity to various researchers focusing on wave approximations.
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Additional resources for Symmetric Discontinuous Galerkin Methods for 1-D Waves: Fourier Analysis, Propagation, Observability and Applications
0/ < 0, eOs;sp . = h/ > 0, and there exists a unique h . 0; = h/ such that eOs;sp given by cos. 15) D = h. 4 are more technical and they will be given in Appendix A. 2. 2 Analysis of the Two Group Velocities 39 • Property (c1) is related to the convergence as h ! 0 of the SIPG approximations to the continuous wave equation, for which the group velocity is identically one. , FD or P1 -FEM). As a consequence of that, there are high-frequency wave packets concentrated on the physical dispersion diagram propagating at arbitrarily low speed.
Chapter 4 Fourier Analysis of the Discontinuous Galerkin Methods The purpose of this chapter is to perform a complete Fourier analysis of the discontinuous Galerkin method under consideration. We obtain explicit formulas for the two classes of eigenvalues and eigenvectors of the Fourier symbol of the discrete Laplacian, the so-called physical and spurious modes. In the last part of this chapter, we analyze fine properties of both Fourier modes as, for example, the behavior of the corresponding group velocities which are first-order derivatives of the square roots of the eigenvalues of the discrete Laplacian.
The proof is similar to the one for the wave equation. 26). Chapter 2 Discontinuous Galerkin Approximations and Main Results The second chapter of the book is twofold. First, we briefly present the approximation schemes under consideration and their main properties. In particular, we introduce the discontinuous Galerkin (DG) semi-discretization of the wave and Klein–Gordon equations using the so-called symmetric interior penalty DG method in its simplest version, in which piecewise linear polynomials are used on uniform meshes.