Mixed virtual volume methods for elliptic problems
- Authors
- Jo, Gwanghyun; Kwak, Do Y.
- Issue Date
- May-2022
- Publisher
- Pergamon Press Ltd.
- Keywords
- Computable L2-projection; Local velocity recovery; Mixed virtual element methods; Mixed virtual volume methods; Nonconforming virtual element methods; Polygonal/polyhedral meshes
- Citation
- Computers and Mathematics with Applications, v.113, pp 345 - 352
- Pages
- 8
- Indexed
- SCIE
SCOPUS
- Journal Title
- Computers and Mathematics with Applications
- Volume
- 113
- Start Page
- 345
- End Page
- 352
- URI
- https://scholarworks.bwise.kr/erica/handle/2021.sw.erica/115248
- DOI
- 10.1016/j.camwa.2022.03.038
- ISSN
- 0898-1221
1873-7668
- Abstract
- We develop a class of mixed virtual volume methods for elliptic problems on polygonal/polyhedral grids. Unlike the mixed virtual element methods introduced in [22,13], our methods are reduced to symmetric, positive definite problems for the primary variable without using Lagrangian multipliers. We start from the usual way of changing the given equation into a mixed system using the Darcy's law, u=−K∇p. By integrating the system of equations with some judiciously chosen test spaces on each element, we define new mixed virtual volume methods of all orders. We show that these new schemes are equivalent to the nonconforming virtual element methods for the primal variable p. Once the primary variable is computed solving the symmetric, positive definite system, all the degrees of freedom for the Darcy velocity are locally computed. Also, the L2-projection onto the polynomial space is easy to compute. Hence our work opens an easy way to compute Darcy velocity on the polygonal/polyhedral grids. For the lowest order case, we give a formula to compute a Raviart-Thomas space like representation which satisfies the conservation law. An optimal error analysis is carried out and numerical results are presented which support the theory. © 2022 Elsevier Ltd
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