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**Parabolic finite volume element equations in nonconvex polygonal domains.**
*(English)*
Zbl 1168.65051

The numerical analysis of the 2D heat equation with homogeneous Dirichlet boundary conditions in a nonconvex polygonal domain is considered. For simplicity, exact one interior angle \(\omega\) of the polygonal domain is reentrant. i.e. \(\omega \in (\pi, 2\pi)\).

The work presents continuation of authors’ previous works concerning a numerical analysis of the heat equation on a convex domain using the finite volume method and error estimates of the heat equation for a nonconvex domain using the finite element method. In this paper, for space discretization the finite volume method is used, more precisely the combination of a primary discretization by triangulation of the domain on which the piecewise linear functions are prescribed and by a dual so called co-volume mesh associated with each vertex of the primary mesh.

An error analysis of the approximate solution given by the Petrov-Galerkin formulation is proved both for semidiscretization in space and fully discretization where for time discretization the backward Euler approximation is used. Due to the singularity of the domain the convergence rate is reduced from optimal \(O(h^2)\) similarly as it was proved in the authors’ previous works for the finite element method to \(O(h^{2\beta}), \;\beta =\frac{\pi}{\omega} \in (\frac12, 1).\) The error estimates in \(H^1\) norm and in the maximum norm are also studied.

Optimal order convergence may be restored by mesh refinement near the corner of the domain.

The work presents continuation of authors’ previous works concerning a numerical analysis of the heat equation on a convex domain using the finite volume method and error estimates of the heat equation for a nonconvex domain using the finite element method. In this paper, for space discretization the finite volume method is used, more precisely the combination of a primary discretization by triangulation of the domain on which the piecewise linear functions are prescribed and by a dual so called co-volume mesh associated with each vertex of the primary mesh.

An error analysis of the approximate solution given by the Petrov-Galerkin formulation is proved both for semidiscretization in space and fully discretization where for time discretization the backward Euler approximation is used. Due to the singularity of the domain the convergence rate is reduced from optimal \(O(h^2)\) similarly as it was proved in the authors’ previous works for the finite element method to \(O(h^{2\beta}), \;\beta =\frac{\pi}{\omega} \in (\frac12, 1).\) The error estimates in \(H^1\) norm and in the maximum norm are also studied.

Optimal order convergence may be restored by mesh refinement near the corner of the domain.

Reviewer: Angela Handlovičová (Bratislava)

### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods 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 element method; parabolic equations; nonconvex polygonal domain; error estimates; convergence; heat equation; semidiscretization; mesh refinement
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\textit{P. Chatzipantelidis} et al., Numer. Methods Partial Differ. Equations 25, No. 3, 507--525 (2009; Zbl 1168.65051)

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### References:

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