A post-processed higher-order multiscale method for nondivergence-form elliptic equations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914480249634816 |
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| author | Hauck, Moritz Maier, Roland Sprekeler, Timo |
| author_facet | Hauck, Moritz Maier, Roland Sprekeler, Timo |
| contents | We study the finite element approximation of linear second-order elliptic partial differential equations in nondivergence form with highly heterogeneous diffusion and drift coefficients. A generalized Cordes condition is imposed to guarantee that a suitably renormalized version of the nondivergence-form differential operator is near the Laplacian. Based on a stabilized symmetric formulation for the gradient that enables the use of $H^1$-conforming approximation spaces, we construct a multiscale method following the methodology of the localized orthogonal decomposition with coarse basis functions tailored to the heterogeneous coefficients. We employ a novel post-processing strategy to obtain higher-order convergence rates, overcoming previous limitations imposed by the low regularity of the load functional. Numerical experiments demonstrate the performance of the method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15144 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A post-processed higher-order multiscale method for nondivergence-form elliptic equations Hauck, Moritz Maier, Roland Sprekeler, Timo Numerical Analysis 65N12, 65N15, 65N30 We study the finite element approximation of linear second-order elliptic partial differential equations in nondivergence form with highly heterogeneous diffusion and drift coefficients. A generalized Cordes condition is imposed to guarantee that a suitably renormalized version of the nondivergence-form differential operator is near the Laplacian. Based on a stabilized symmetric formulation for the gradient that enables the use of $H^1$-conforming approximation spaces, we construct a multiscale method following the methodology of the localized orthogonal decomposition with coarse basis functions tailored to the heterogeneous coefficients. We employ a novel post-processing strategy to obtain higher-order convergence rates, overcoming previous limitations imposed by the low regularity of the load functional. Numerical experiments demonstrate the performance of the method. |
| title | A post-processed higher-order multiscale method for nondivergence-form elliptic equations |
| topic | Numerical Analysis 65N12, 65N15, 65N30 |
| url | https://arxiv.org/abs/2604.15144 |