A construction of higher-order finite volume methods

A New Class of Higher Order Mixed Finite Volume Methods for Elliptic Problems

We introduce a new class of higher order mixed finite volume methods for elliptic problems. We start from the usual way of changing the given equation into a mixed system using the Darcy’s law, u = −K∇p. By integrating the system of equations with some judiciously chosen test spaces on each element, we define new mixed finite volume methods of higher order. We show that these new schemes are eq...

A Posteriori Error Estimates for Higher Order Godunov Finite Volume Methods on Unstructured Meshes

A posteriori error estimates for high order Godunov finite volume methods are presented which exploit the two solution representations inherent in the method, viz. as piecewise constants u0 and cellwise p-th order reconstructed functions R 0 pu0. Using standard duality arguments, we construct exact error representation formulas for derived functionals that are tailored to the class of high orde...

Unified analysis of higher-order finite volume methods for parabolic problems on quadrilateral meshes

Finite volume methods (FVMs) have been widely used in scientific computing and engineering due to their easy implementation and the local conservation property. Lower-order FVMs are tightly related to finite difference or finite element methods, and have been extensively studied for a long time; see, e.g., Angelini et al. (2013), Bank & Rose (1987), Chatzipantelidis et al. (2008), Chou & Ye (20...

Development and Analysis of Higher Order Finite Volume Methods over Rectangles for Elliptic Equations

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 approp...

Development and Analysis of Higher Order Finite Volume Methods over Re tangles for Ellipti Equations