3. A space-time discontinuous Petrov–Galerkin method for acoustic waves
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Johannes Ernesti
Abstract
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 well-posed. 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.
Abstract
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 well-posed. 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.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents IX
- 1. Space-time boundary element methods for the heat equation 1
- 2. Parallel adaptive discontinuous Galerkin discretizations in space and time for linear elastic and acoustic waves 61
- 3. A space-time discontinuous Petrov–Galerkin method for acoustic waves 89
- 4. A space-time DPG method for the wave equation in multiple dimensions 117
- 5. Adaptive space-time isogeometric analysis for parabolic evolution problems 141
- 6. Generating admissible space-time meshes for moving domains in (d + 1) dimensions 185
- 7. Space-time finite element methods for parabolic evolution equations: discretization, a posteriori error estimation, adaptivity and solution 207
- Index 249
- Radon Series on Computational and Applied Mathematics 251
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents IX
- 1. Space-time boundary element methods for the heat equation 1
- 2. Parallel adaptive discontinuous Galerkin discretizations in space and time for linear elastic and acoustic waves 61
- 3. A space-time discontinuous Petrov–Galerkin method for acoustic waves 89
- 4. A space-time DPG method for the wave equation in multiple dimensions 117
- 5. Adaptive space-time isogeometric analysis for parabolic evolution problems 141
- 6. Generating admissible space-time meshes for moving domains in (d + 1) dimensions 185
- 7. Space-time finite element methods for parabolic evolution equations: discretization, a posteriori error estimation, adaptivity and solution 207
- Index 249
- Radon Series on Computational and Applied Mathematics 251
