From simplex to mixed element: Extension of a vertex-centered discretization, focus on accuracy analysis and 3D RANS applications

Abstract

Standard unstructured-grid CFD simulations generally rely on a cell-centered Finite Volume discretization applied to mixed-element grids. The interest in such approach is using elements that are aligned along a privileged direction in the region close to the boundary, and at the same time unstructured elements near complex geometrical details or in farfield regions. This paper proposes a novel version of the mixed Finite Element/Finite Volume approximation (Debiez and Dervieux 2000), which is a vertex-centered method known to produce second-order accurate solutions even on highly anisotropic adapted meshes composed of simplex elements (i.e., triangles and tetrahedra) (Alauzet and Loseille, 2010; Barral et al., 2017; Alauzet et al., 2018; Belme et al., 2019). The extension of this approach for two-dimensional mixed-element meshes was proposed in Tarsia Morisco et al. (2024) and involves the APproximated Finite Element -APFE- method (Puigt et al., 2010) to discretize diffusion. In this work we make the definitive step forward to handle three-dimensional mixed-element meshes: designing a second-order accurate scheme for smooth meshes involving tetrahedra, prisms and pyramids.

The present work focuses on two key aspects. One concerns the 3D extension of the APFE method. A detailed error analysis of this vertex-centered approach is provided for prisms and pyramids. The second ingredient deals with an innovative algorithm to compute the truncation error for linear problems. In contrast to usual methods, the one proposed here permits to compute exactly the coefficients related to each terms of error for any mesh, and can be implemented in any solver with a low development effort.