5. Adaptive space-time isogeometric analysis for parabolic evolution problems
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Ulrich Langer
Abstract
The paper proposes new locally stabilized space-time isogeometric analysis approximations to initial boundary value problems of the parabolic type. Previously, similar schemes (but weighted with a global mesh parameter) have been presented and studied by U. Langer, M. Neumüller, and S. Moore (2016). The current work devises a localized version of this scheme, which is suited for adaptive mesh refinement. We establish coercivity, boundedness, and consistency of the corresponding bilinear form. Using these fundamental properties together with standard approximation error estimates for B-splines and NURBS, we show that the space-time isogeometric analysis solutions generated by the new scheme satisfy asymptotically optimal a priori discretization error estimates. Error indicators used for mesh refinement are based on a posteriori error estimates of the functional type that have been introduced by S. Repin (2002), and later rigorously studied in the context of isogeometric analysis by U. Langer, S. Matculevich, and S. Repin (2017). Numerical results discussed in the paper illustrate an improved convergence of global approximation errors and respective error majorants. They also confirm the local efficiency of the error indicators produced by the error majorants.
Abstract
The paper proposes new locally stabilized space-time isogeometric analysis approximations to initial boundary value problems of the parabolic type. Previously, similar schemes (but weighted with a global mesh parameter) have been presented and studied by U. Langer, M. Neumüller, and S. Moore (2016). The current work devises a localized version of this scheme, which is suited for adaptive mesh refinement. We establish coercivity, boundedness, and consistency of the corresponding bilinear form. Using these fundamental properties together with standard approximation error estimates for B-splines and NURBS, we show that the space-time isogeometric analysis solutions generated by the new scheme satisfy asymptotically optimal a priori discretization error estimates. Error indicators used for mesh refinement are based on a posteriori error estimates of the functional type that have been introduced by S. Repin (2002), and later rigorously studied in the context of isogeometric analysis by U. Langer, S. Matculevich, and S. Repin (2017). Numerical results discussed in the paper illustrate an improved convergence of global approximation errors and respective error majorants. They also confirm the local efficiency of the error indicators produced by the error majorants.
Chapters in this book
- 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
Chapters in this book
- 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
