Computational multiscale methods for nondivergence-form elliptic partial differential equations
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909236975370240 |
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| author | Freese, Philip Gallistl, Dietmar Peterseim, Daniel Sprekeler, Timo |
| author_facet | Freese, Philip Gallistl, Dietmar Peterseim, Daniel Sprekeler, Timo |
| contents | This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence-form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence-form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_13731 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Computational multiscale methods for nondivergence-form elliptic partial differential equations Freese, Philip Gallistl, Dietmar Peterseim, Daniel Sprekeler, Timo Numerical Analysis 35J15, 65N12, 65N30 This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence-form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence-form. |
| title | Computational multiscale methods for nondivergence-form elliptic partial differential equations |
| topic | Numerical Analysis 35J15, 65N12, 65N30 |
| url | https://arxiv.org/abs/2211.13731 |