Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes. (English) Zbl 1274.65251

Bradji, Abdallah; Fuhrmann, Jürgen

Appl. Math., Praha 58, No. 1, 1-38 (2013).

The article is concerned with the derivation of error estimates for a finite volume scheme applied to the nonstationary heat equation with homogeneous boundary conditions. The space-time discretization employs the discrete gradient construction from the so-called SUSHI scheme of R. Eymard et al. [IMA J. Numer. Anal. 30, 1009–1043 (2010; Zbl 1202.65144)], originally applied to stationary elliptic problems. Such a scheme can be applied and analyzed on very general nonconforming meshes with star-shaped polyhedral elements. Error estimates in the \(L^{\infty }(H^1_0)\) and \(W^{1,\infty }(L^2)\) norms are derived. The proof uses estimates of the discrete gradient and properties of the involved discrete spaces from [loc. cit.]. Moreover, a suitable auxiliary problem is introduced, for which error estimates are known, and the final results of the paper are derived by comparison of the auxiliary problem and the considered finite volume scheme. The error estimates are derived under rather strong regularity assumptions on the exact solution.

Reviewer: Václav Kučera (Praha)

MSC:

65M08	Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
35K05	Heat equation
65M50	Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs

Keywords:

finite volume method; heat equation; error estimate; discrete gradient; nonconforming grids; star-shaped elements; SUSHI scheme

Citations:

Zbl 1202.65144

References:

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