Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems

Abstract

Abstract. We construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order PDEs in the state variable (e.g., pressure) and its flux (e.g., Darcy velocity). An important point is that no staggered grids or covolumes are used to stabilize the system. Only a single primary grid system is adopted, and the degrees of freedom are imposed on the interfaces. The approximate flux is sought in the lowest-order Raviart-Thomas space and the pressure field in the rotated-Q1 nonconforming space. Furthermore, we demonstrate that the present finite volume method can be interpreted as a rotated-Q1 nonconforming finite element method for the pressure with a simple local recovery of flux. Numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method.

A second-order finite volume element method on quadrilateral meshes for elliptic equations

“…Here p * and p * are positive constants. Finite volume element (FVE) methods [3,6,7,16,17], also named as generalized difference methods [13,14,19,22] or box methods [1,11], have been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer or petroleum engineering. The FVE methods involve two spaces.…”

“…For some details we refer to the articles on triangular meshes [3,16] or on quadrilateral meshes [6,13]. In present time, one approach to obtain higher order schemes is using uniform or symmetric meshes to obtain superconvergence.…”

A posteriori error analysis for locally conservative mixed methods

“…The most popular one among such mixed methods is the mixed finite element method; see, for example, [8,18] for an extensive discussion on this topic (and the references therein). There are also finite volume type methods such as mixed covolume methods [10,12] and mixed finite volume methods on nonstaggered grids [13,16].…”

“…Recently, it was revealed in [13] that the discretization (5.7) can be interpreted as a mixed finite volume method if we replace f and g N byf and Q 0 h g N , respectively. More specifically, the modified method of (5.7) (5.8)…”

New mixed finite volume methods for second order eliptic problems

“…[12,[17][18][19]). A mixed finite volume method on non-staggered triangular grids was proposed by Courbet and Croisille [23] and later extended to quadrilateral grids by Chou, Kwak and Kim [20]. It has many distinct features when compared with the two methods above, like violation of the inf-sup condition and no use of covolumes.…”