A face‐centred finite volume method for second‐order elliptic problems

Abstract

Summary This work proposes a novel finite volume paradigm, ie, the face‐centred finite volume (FCFV) method. Contrary to the popular vertex and cell‐centred finite volume methods, the novel FCFV defines the solution on the mesh faces (edges in two dimensions) to construct locally conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin formulation with constant degree of approximation, and thus inheriting the convergence properties of the classical hybridisable discontinuous Galerkin. The resulting FCFV features a global problem in terms of a piecewise constant function defined on the faces of the mesh. The solution and its gradient in each element are then recovered by solving a set of independent element‐by‐element problems. The mathematical formulation of FCFV for Poisson and Stokes equation is derived, and numerical evidence of optimal convergence in two dimensions and three dimensions is provided. Numerical examples are presented to illustrate the accuracy, efficiency, and robustness of the proposed methodology. The results show that, contrary to other finite volume methods, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. In addition, the FCFV method shows its better performance, accuracy, and robustness using simplicial elements, facilitating its application to problems involving complex geometries in three dimensions.

A locking-free face-centred finite volume (FCFV) method for linear elastostatics

“…Recently, a face-centred finite volume (FCFV) method which defines the unknowns over the faces of the mesh elements has been introduced for Poisson and Stokes problems [36]. In the present work, the FCFV method is extended to simulate the behaviour of deformable bodies under the assumption of small displacements.…”

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

“…122 Starting from the framework discussed above, a novel efficient and robust finite volume (FV) rationale has been proposed in. 106,107 In order to describe this approach, an incompressible Stokes flow is considered…”

“…FCFV solves the problem in two phases. 106,107 First, by applying the divergence theorem to Equation (20) and exploiting the definition of the numerical normal flux on the boundary in Equation (23), a set of n el local integral problems is obtained…”

“…The FCFV global problem has a sparse block structure allowing a computationally efficient implementation, see. 106 Moreover, FCFV local computations to determine velocity, pressure and gradient of velocity in the centroid of each cell solely involve elementary operations cell-by-cell for which modern parallel architectures can be exploited. FCFV inherits the approximation properties of the corresponding high-order HDG formulation from which it is derived.…”

A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

“…Therefore, the mass conservation is satisfied within each element and the summation of these discrete equations can be exactly reduced to the domain boundary. More recently, Sevilla et al pointed out that the face‐centered finite volume method shows better performance, accuracy, and robustness involving complex geometries in three dimensions. Oefelein and Sankaran implemented the same formulation in generalized curvilinear coordinates for the large eddy simulation of the compressible turbulent multiphase combustion processes as the next generation turbulent combustion code (RAPTOR code) on massively parallel supercomputers.…”