Mixed Covolume Methods for Elliptic Problems on Triangular Grids

Abstract

We consider a covolume or finite volume method for a system of first-order PDEs resulting from the mixed formulation of the variable coefficient-matrix Poisson equation with the Neumann boundary condition. The system may represent either the Darcy law and the mass conservation law in anisotropic porous media flow, or Fourier law and energy conservation. The velocity and pressure are approximated by the lowest order Raviart-Thomas space on triangles. We prove its first-order optimal rate of convergence for the approximate velocities in the L 2 -and H(div; Ω)-norms as well as for the approximate pressures in the L 2 -norm. Numerical experiments are included.

Error estimates in $L^2$, $H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach

“…Comparison of the two systems then often leads to fruitful analysis. (This and similar ideas have been successfully exploited in [6,12,11,8,9,10].) Now if one strives to carry out this program, one is very naturally led into considering the quantity (d for "deviation")…”

Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems

“…It was later extended and analyzed in a more effective way by Chou and Tang [15] to general tensor-coefficient problems. Furthermore, following the previous successful and consistent viewpoint [11,12,16] that any given finite volume method should be related to a close finite element method, they provided in [14,15], among other things, a new framework showing equivalence between mixed finite volume methods and nonconforming Galerkin methods with a cheap local recovery of the fluxes. The significance of this is that, unlike lower order mixed finite element methods, mixed finite volume methods can decouple the pressure from the flux and compute it basically cost free.…”

“…On the other hand, the finite volume method has been also applied to the system (1.2) in several ways. For example, see [6,11,12,25] for mixed covolume methods, and [35,36,38] for different approaches.…”

“…In the mixed finite volume literature, it is common to introduce a secondary grid or dual grid of covolumes to enforce equality between the numbers of equations and unknowns [6,11,12,13,38]. Thus it is rather surprising that there exist finite volume methods such as those in [14,15,17] achieving stability using a single grid system while violating the inf-sup condition on the pressure-velocity spaces.…”

Modified Method of Characteristics Combined with Finite Volume Element Methods for Incompressible Miscible Displacement Problems in Porous Media

“…This transfer operator plays a vital role in deriving the optimal error estimates. Earlier, Chou et al [13,14] have discussed and analyzed mixed covolume or FVEM for the second-order linear elliptic problems in two-dimensional domains. The standard FVEM can also be considered as a Petrov-Galerkin finite element method in which the trial space is chosen as 0 piecewise linear polynomials on the finite element partition of the domain and the test space, as piecewise constants over the control volumes are to be defined in Section 2.…”