Development and analysis of higher order finite volume methods over rectangles for elliptic equations
The paper presents a systematic way to derive higher order finite volume methods (FVM) for elliptic equations. The underlying domain must allow a triangulation into rectangles. Starting point of the derivation is a mixed finite element method (FEM) and its crucial step consists in approximating a certain inner product by a quadrature rule such that the matrix which arises from applying the quadrature rule is diagonal and the numerical integration does not decrease the order of convergence of the original mixed FEM. In this way, equations in a scalar variable are obtained which define the FVM. The paper studies in detail this approach for the Brezzi-Douglas-Fortin-Marini element of index two (BDFM\(_2\)). Optimal error estimates for the scalar variable and the recovered flux are proved by considering the difference of the FVM solution to the corresponding mixed FEM solution. An analogous analysis leading to optimal error estimates is carried out for a FVM based on the Raviart-Thomas-Nedelec mixed finite element of arbitrary order in 2D and 3D. A FVM based on BDFM\(_2\) in 3D is derived and optimal error estimates are proved provided that the diffusion matrix is constant on each mesh cell. Last, it is discussed that a satisfactory FVM based on BDFM\(_3\) currently cannot be derived. Numerical examples employing the proposed FVM are not presented in this paper.
Raviart-Thomas-Nedelec mixed finite element
scientific article; zbMATH DE number 1903987