Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients
The paper develops the essentially optimal sparse tensor product finite element method for solving two scale elliptic and parabolic problems in a domain D⊂Rd, d=2,3, which is embedded with a periodic array of inclusions of microscopic sizes and spacing. The two scale coefficient is thus discontinuou...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2025
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| Online Access: | https://hdl.handle.net/10356/182626 |
| _version_ | 1826126363471380480 |
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| author | Pang, Chen Hui Hoang, Viet Ha |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Pang, Chen Hui Hoang, Viet Ha |
| author_sort | Pang, Chen Hui |
| collection | NTU |
| description | The paper develops the essentially optimal sparse tensor product finite element method for solving two scale elliptic and parabolic problems in a domain D⊂Rd, d=2,3, which is embedded with a periodic array of inclusions of microscopic sizes and spacing. The two scale coefficient is thus discontinuous in the fast variable. We obtain approximations for the solution of the homogenized equation and the scale interaction term, i.e. all the macroscopic and microscopic information, within a prescribed level of accuracy, using only an essentially optimal number of degrees of freedom, which is equal (apart from a possible logarithmic factor) to that required to solve one macroscopic scale problem in D. This is achieved by solving the two scale homogenized problem, and utilizing the regularity of the scale interaction term in all the slow and fast variables at the same time. However, unlike problems considered in the literature (e.g. Hoang and Schwab, 2004/05 [16]), the scale interaction term is only piecewise regular in the fast variable. We employ the discretization schemes developed for interface problems (Chen and Zou, 1998 [6], and Li et al., 2010 [20]) for the fast variable. Numerical correctors are developed from the finite element solutions with errors in terms of the finite element mesh size and the microscopic scale. Numerical examples that verify the theoretical convergence rates of the sparse tensor product finite elements are presented. |
| first_indexed | 2025-03-09T14:16:51Z |
| format | Journal Article |
| id | ntu-10356/182626 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2025-03-09T14:16:51Z |
| publishDate | 2025 |
| record_format | dspace |
| spelling | ntu-10356/1826262025-02-12T02:21:16Z Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients Pang, Chen Hui Hoang, Viet Ha School of Physical and Mathematical Sciences Mathematical Sciences Discontinuous coefficients Element method The paper develops the essentially optimal sparse tensor product finite element method for solving two scale elliptic and parabolic problems in a domain D⊂Rd, d=2,3, which is embedded with a periodic array of inclusions of microscopic sizes and spacing. The two scale coefficient is thus discontinuous in the fast variable. We obtain approximations for the solution of the homogenized equation and the scale interaction term, i.e. all the macroscopic and microscopic information, within a prescribed level of accuracy, using only an essentially optimal number of degrees of freedom, which is equal (apart from a possible logarithmic factor) to that required to solve one macroscopic scale problem in D. This is achieved by solving the two scale homogenized problem, and utilizing the regularity of the scale interaction term in all the slow and fast variables at the same time. However, unlike problems considered in the literature (e.g. Hoang and Schwab, 2004/05 [16]), the scale interaction term is only piecewise regular in the fast variable. We employ the discretization schemes developed for interface problems (Chen and Zou, 1998 [6], and Li et al., 2010 [20]) for the fast variable. Numerical correctors are developed from the finite element solutions with errors in terms of the finite element mesh size and the microscopic scale. Numerical examples that verify the theoretical convergence rates of the sparse tensor product finite elements are presented. Ministry of Education (MOE) Nanyang Technological University The authors gratefully acknowledge the financial support of a Postgraduate Scholarship of Nanyang Technological University, and the Tier 2 grant T2EP20123-0047 awarded by the Singapore Ministry of Education. 2025-02-12T02:21:16Z 2025-02-12T02:21:16Z 2025 Journal Article Pang, C. H. & Hoang, V. H. (2025). Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients. Computers and Mathematics With Applications, 179, 17-40. https://dx.doi.org/10.1016/j.camwa.2024.11.018 0898-1221 https://hdl.handle.net/10356/182626 10.1016/j.camwa.2024.11.018 2-s2.0-85211062286 179 17 40 en T2EP20123-0047 Computers and Mathematics with Applications © 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies. |
| spellingShingle | Mathematical Sciences Discontinuous coefficients Element method Pang, Chen Hui Hoang, Viet Ha Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients |
| title | Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients |
| title_full | Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients |
| title_fullStr | Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients |
| title_full_unstemmed | Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients |
| title_short | Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients |
| title_sort | sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients |
| topic | Mathematical Sciences Discontinuous coefficients Element method |
| url | https://hdl.handle.net/10356/182626 |
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