Difference schemes for parabolic equations on triangular grids.

*(English. Russian original)*Zbl 1074.65103
Russ. Math. 47, No. 1, 51-57 (2003); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2003, No. 1, 53-59 (2003).

From the introduction: For the construction of difference schemes on unstructured triangular grids, we consider the boundary value problem for the two-dimensional equation of heat conductivity. Such a choice of a model problem is explained by the fact that the equation of heat conductivity includes two basic differential operators, i.e., the divergence and the gradient, which enter into all basic equations of the mechanics of a solid medium. The investigation of the properties of difference analogs of these operators on unstructured grids makes it possible to generalize in the future both theoretical and practical results onto other types of partial equations.

We consider the method of constructing conservative difference schemes of the second and higher orders of accuracy. The schemes of the second-order are constructed for arbitrary (including defective ones) grids failing to satisfy the Delone criterion. These grids can be used in the technological calculation. The schemes of an augmented order of accuracy are oriented to triangular grids which are close to uniform ones.

We consider the method of constructing conservative difference schemes of the second and higher orders of accuracy. The schemes of the second-order are constructed for arbitrary (including defective ones) grids failing to satisfy the Delone criterion. These grids can be used in the technological calculation. The schemes of an augmented order of accuracy are oriented to triangular grids which are close to uniform ones.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |