Volume 20, Issue 1
The Mixed Finite Volume Methods for Stokes Problem Based on MINI Element Pair

Int. J. Numer. Anal. Mod., 20 (2023), pp. 134-151.

Published online: 2022-11

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• Abstract

In this paper, we present and analyze MINI Mixed finite volume element methods (MINI-FVEM) for Stokes problem on triangular meshes. The trial spaces for velocity and pressure are chosen as MINI element pair, and the test spaces for velocity and pressure are taken as the piecewise constant function spaces on the respective dual grid. It is worth noting that the bilinear form derived from the gradient operator and the bilinear form derived from the divergence are unsymmetric. With the help of two new transformation operators, we establish the equivalence of bilinear forms for gradient operator between finite volume methods and finite element methods, and the equivalence of bilinear forms for divergence operator between finite volume methods and finite element methods, so the inf-sup conditions are obtained. By the element analysis methods, we give the positive definiteness of bilinear form for Laplacian operator. Based on the stability, convergence analysis of schemes are established. Numerical experiments are presented to illustrate the theoretical results.

• AMS Subject Headings

65N08, 76D07, 76M12

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@Article{IJNAM-20-134, author = {Yang , Hongtao and Li , Yonghai}, title = {The Mixed Finite Volume Methods for Stokes Problem Based on MINI Element Pair}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2022}, volume = {20}, number = {1}, pages = {134--151}, abstract = {

In this paper, we present and analyze MINI Mixed finite volume element methods (MINI-FVEM) for Stokes problem on triangular meshes. The trial spaces for velocity and pressure are chosen as MINI element pair, and the test spaces for velocity and pressure are taken as the piecewise constant function spaces on the respective dual grid. It is worth noting that the bilinear form derived from the gradient operator and the bilinear form derived from the divergence are unsymmetric. With the help of two new transformation operators, we establish the equivalence of bilinear forms for gradient operator between finite volume methods and finite element methods, and the equivalence of bilinear forms for divergence operator between finite volume methods and finite element methods, so the inf-sup conditions are obtained. By the element analysis methods, we give the positive definiteness of bilinear form for Laplacian operator. Based on the stability, convergence analysis of schemes are established. Numerical experiments are presented to illustrate the theoretical results.

}, issn = {2617-8710}, doi = {https://doi.org/ 10.4208/ijnam2023-1006}, url = {http://global-sci.org/intro/article_detail/ijnam/21207.html} }
TY - JOUR T1 - The Mixed Finite Volume Methods for Stokes Problem Based on MINI Element Pair AU - Yang , Hongtao AU - Li , Yonghai JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 134 EP - 151 PY - 2022 DA - 2022/11 SN - 20 DO - http://doi.org/ 10.4208/ijnam2023-1006 UR - https://global-sci.org/intro/article_detail/ijnam/21207.html KW - Stokes problem, MINI element, mixed finite volume methods, inf-sup condition. AB -

In this paper, we present and analyze MINI Mixed finite volume element methods (MINI-FVEM) for Stokes problem on triangular meshes. The trial spaces for velocity and pressure are chosen as MINI element pair, and the test spaces for velocity and pressure are taken as the piecewise constant function spaces on the respective dual grid. It is worth noting that the bilinear form derived from the gradient operator and the bilinear form derived from the divergence are unsymmetric. With the help of two new transformation operators, we establish the equivalence of bilinear forms for gradient operator between finite volume methods and finite element methods, and the equivalence of bilinear forms for divergence operator between finite volume methods and finite element methods, so the inf-sup conditions are obtained. By the element analysis methods, we give the positive definiteness of bilinear form for Laplacian operator. Based on the stability, convergence analysis of schemes are established. Numerical experiments are presented to illustrate the theoretical results.

Hongtao Yang & Yonghai Li. (2022). The Mixed Finite Volume Methods for Stokes Problem Based on MINI Element Pair. International Journal of Numerical Analysis and Modeling. 20 (1). 134-151. doi: 10.4208/ijnam2023-1006
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