Combined finite element–finite volume method (convergence analysis).

*(English)*Zbl 0938.65111The author presents a numerical method for a scalar nonlinear convection-diffusion problem and performs its theoretical analysis. The method consists in combining the finite element method and the finite volume method in an appropriate way. In fact, the author combines the \(P_1\)-conforming finite element method for discretizing the diffusion terms with an upstream discretization of the nonlinear convective term on a “finite volume” mesh dual to the FEM triangular grid. This upwind discretization takes into account the dominating influence of the convective term in the case of higher Reynolds numbers, and it is viewed as a finite volume discretization.

The continuous problem is described at the beginning of the paper and its weak solution is defined. Then the corresponding discrete problem is formulated and its properties are studied. The main result of the paper is the proof of the strong convergence in \(L^2\)-norm of the numerical solution to the exact weak solution of the continuous problem under consideration.

A similar problem was studied by the author, M. Feistauer and J. Felcman before [Numer. Methods Partial Differ. Equations 13, 163-190 (1997; Zbl 0869.65057)]. In the present paper, the author needs less regularity of the initial data and proves the convergence theorem without the assumption that the triangulation is of a weakly acute type. On the other hand, it is now necessary to assume that the initial data is small in a certain sense.

The continuous problem is described at the beginning of the paper and its weak solution is defined. Then the corresponding discrete problem is formulated and its properties are studied. The main result of the paper is the proof of the strong convergence in \(L^2\)-norm of the numerical solution to the exact weak solution of the continuous problem under consideration.

A similar problem was studied by the author, M. Feistauer and J. Felcman before [Numer. Methods Partial Differ. Equations 13, 163-190 (1997; Zbl 0869.65057)]. In the present paper, the author needs less regularity of the initial data and proves the convergence theorem without the assumption that the triangulation is of a weakly acute type. On the other hand, it is now necessary to assume that the initial data is small in a certain sense.

Reviewer: P.Přikryl (Praha)

##### MSC:

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

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

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |