4. A space-time DPG method for the wave equation in multiple dimensions
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Jay Gopalakrishnan
Abstract
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 wellposedness 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 presented.
Abstract
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 wellposedness 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 presented.
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
