New mixed finite volume methods for second order eliptic problems

Kim, Kwang Y.

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 123-147.

In this paper we introduce and analyze new mixed finite volume methods for second order elliptic problems which are based on $H\left(\mathrm{div}\right)$-conforming approximations for the vector variable and discontinuous approximations for the scalar variable. The discretization is fulfilled by combining the ideas of the traditional finite volume box method and the local discontinuous Galerkin method. We propose two different types of methods, called Methods I and II, and show that they have distinct advantages over the mixed methods used previously. In particular, a clever elimination of the vector variable leads to a primal formulation for the scalar variable which closely resembles discontinuous finite element methods. We establish error estimates for these methods that are optimal for the scalar variable in both methods and for the vector variable in Method II.

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