A unified analysis of a class of quadratic finite volume element schemes on triangular meshes (Q2200078)

The authors provide a general framework for the coercivity analysis of a class of quadratic FVE (Finite Volume Element) schemes on triangular meshes. This class of schemes have two parameters \(\alpha \in (0, 1/2)\) and \(\beta \in (0, 2/3)\), which cover all the existing quadratic schemes of Lagrange type. Using an element analysis and a new mapping from the trial function space to the test function space, it is proved that each element matrix can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, and the third part can be expressed as the tensor product of two vectors. Thanks to this result, it is obtained a sufficient condition to guarantee the existence, uniqueness, and coercivity result of the FVE solution on triangular meshes. Using this sufficient condition, some minimum angle conditions are derived and they depend only on the constructive parameters of the schemes. Thanks to these results, some existing coercivity results are improved. An optimal \(H^1\) error estimate is also proved.