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Validation of a 2D cell-centered finite volume method for elliptic equations. (English) Zbl 07316740

Summary: Following the approach in [the first author and R. Temam, “Cell centered finite volume methods using Taylor series expansion scheme without fictitious domains”, Int. J. Numer. Anal. Mod. 7, 1–29 (2010); “Convergence of a cell-centered finite volume method and application to elliptic equations”, ibid. 12, 536–566 (2015)], we construct the Finite Volume (FV) approximations of a class of elliptic equations and perform numerical computations where a 2D domain is discretized by convex quadrilateral meshes. The FV method with Taylor Series Expansion Scheme (TSES), which is properly adjusted from a version widely used in engineering, is tested in a box, annulus, and in a domain which includes a topography at the bottom boundary. By comparing with other related convergent FV schemes in [Z. Sheng and G. Yuan, Transp. Theory Stat. Phys. 37, No. 2–4, 171–207 (2008; Zbl 1375.82100); I. Aavatsmark, Comput. Geosci. 6, No. 3–4, 405–432 (2002; Zbl 1094.76550); F. Hermeline, J. Comput. Phys. 160, No. 2, 481–499 (2000; Zbl 0949.65101); S. Faure et al., Int. J. Comput. Math. 93, No. 10, 1787–1799 (2016; Zbl 1356.65232)], we numerically verify that our FV method is a convergent 2nd order scheme that manages well the complex geometry. The advantage of our scheme is on its simple structure which do not require any special reconstruction of dual type mesh for computing the nodal approximations or discrete gradients.

MSC:

65-XX Numerical analysis
65Nxx Numerical methods for partial differential equations, boundary value problems
76-XX Fluid mechanics
76Mxx Basic methods in fluid mechanics
65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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