Convection-adapted BEM-based FEM
This work addresses the need for stable discretization methods for convection-dominated problems in computational fluid dynamics and related fields, offering an incremental improvement over existing BEM-based FEM approaches.
The paper introduces a new discretization method for 3D convection-diffusion-reaction problems that uses PDE-harmonic shape functions on polyhedral elements, constructed via local boundary element techniques. The method improves stability for convection-dominated problems compared to standard FEM and previous BEM-based FEM approaches, as shown in numerical experiments.
We present a new discretization method for homogeneous convection-diffusion-reaction boundary value problems in 3D that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments.