Approximating inverse FEM matrices on non-uniform meshes with $\mathcal{H}$-matrices
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866929433454051328 |
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| author | Angleitner, Niklas Faustmann, Markus Melenk, Jens Markus |
| author_facet | Angleitner, Niklas Faustmann, Markus Melenk, Jens Markus |
| contents | We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse $\mathcal{H}$-matrix format. For a large class of shape regular but possibly non-uniform meshes including graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the $\mathcal{H}$-matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_04999 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Approximating inverse FEM matrices on non-uniform meshes with $\mathcal{H}$-matrices Angleitner, Niklas Faustmann, Markus Melenk, Jens Markus Numerical Analysis We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse $\mathcal{H}$-matrix format. For a large class of shape regular but possibly non-uniform meshes including graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the $\mathcal{H}$-matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers. |
| title | Approximating inverse FEM matrices on non-uniform meshes with $\mathcal{H}$-matrices |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2005.04999 |