Nodal AMG Coarsening and Interpolation for PDE Systems
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866929688144773120 |
|---|---|
| author | Brannick, James Falgout, Robert Kahl, Karsten Schroder, Jacob Shen, Taoli |
| author_facet | Brannick, James Falgout, Robert Kahl, Karsten Schroder, Jacob Shen, Taoli |
| contents | We present an approach to constructing a practical coarsening algorithm and interpolation operator for the algebraic multigrid (AMG) method, tailored towards systems of partial differential equations (PDEs) with large near-kernels, such as H(curl) and H(div). Our method builds on compatible relaxation (CR) and the ideal interpolation model within the generalized AMG (GAMG) framework but introduces several modifications to define an AMG method for PDE systems. We construct an interpolation operator through a coarsening process that first coarsens a nodal dual problem and then builds the coarse and fine variables using a matching algorithm. Our interpolation follows the ideal formulation; however, we enhance the sparsity of ideal interpolation by decoupling the fine and coarse variables completely. When the coarse variables align with the geometric refinement, our method reproduces re-discretization on unstructured meshes. Together with an automatic smoother construction scheme that identifies the local near kernels, our approach forms a complete two-grid method. Finally, we also show numerical results that demonstrate the effectiveness of this interpolation scheme by applying it to targeted problems and the Stokes system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nodal AMG Coarsening and Interpolation for PDE Systems Brannick, James Falgout, Robert Kahl, Karsten Schroder, Jacob Shen, Taoli Numerical Analysis We present an approach to constructing a practical coarsening algorithm and interpolation operator for the algebraic multigrid (AMG) method, tailored towards systems of partial differential equations (PDEs) with large near-kernels, such as H(curl) and H(div). Our method builds on compatible relaxation (CR) and the ideal interpolation model within the generalized AMG (GAMG) framework but introduces several modifications to define an AMG method for PDE systems. We construct an interpolation operator through a coarsening process that first coarsens a nodal dual problem and then builds the coarse and fine variables using a matching algorithm. Our interpolation follows the ideal formulation; however, we enhance the sparsity of ideal interpolation by decoupling the fine and coarse variables completely. When the coarse variables align with the geometric refinement, our method reproduces re-discretization on unstructured meshes. Together with an automatic smoother construction scheme that identifies the local near kernels, our approach forms a complete two-grid method. Finally, we also show numerical results that demonstrate the effectiveness of this interpolation scheme by applying it to targeted problems and the Stokes system. |
| title | Nodal AMG Coarsening and Interpolation for PDE Systems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2501.16248 |