Advances in Computational Mathematics 19: 3–33, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands. Development and analysis of higher order finite volume

Currently used finite volume methods are essentially low order methods. In this paper, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. In particular, we restrict ourselves to finite volume methods posed over rectangular partitions and begin by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of in-dex two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we de-rive finite volume methods of all orders from the Raviart–Thomas two-dimensional rectangu-lar elements; we also find finite volume methods to associate with BDFM2 three-dimensional rectangles. In each case, we obtain optimal error estimates for both the scalar variable and the recovered flux