This paper presents the relationships between some numerical methods suitable for a heterogeneous elliptic equation with application to reservoir simulation. The methods discussed are the classical mixed finite element method (MFEM), the control-volume mixed finite element method (CVMFEM), the support operators method (SOM), the enhanced cell-centered finite difference method (ECCFDM), and the ...

The immersed boundary (IB) method is an approach to problems of fluid-structure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some versions of the IB method can suffer from poor volume conservation. Methods ...

In the present work we introduce a new family of cell-centered Finite Volume schemes for anisotropic and heterogeneous diffusion operators inspired by the MPFA L method. A very general framework for the convergence study of finite volume methods is provided and then used to establish the convergence of the new method. Fairly general meshes are covered and a computable sufficient criterion for c...

We extend a result of Nicaise [13] for the a posteriori error estimation of the cell-centered finite volume method for the numerical solution of elliptic problems. Having computed the piecewise constant finite volume solution uh, we compute a Morley-type interpolant Iuh. For the exact solution u, the energy error ‖∇T (u− Iuh)‖L2 can be controlled efficiently and reliably by a residual-based a p...

We have successfully extended our implicit hybrid finite element/volume solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered finite volume method. The pre...

Recently, a vertex-based spatial reconstruction for unstructured cell-centered finite volume method has been proposed, showing advantages in accuracy, convergence, and efficiency. However, this was only applied inviscid flows. Moreover, shock-capturing computations, the conventional cell-based limiter choice to suppress numerical oscillations. In work, is extended solution of viscous flows, whi...