Convection-adapted BEM-based FEM

Abstract

We present a new discretization method for homogeneous convectiondiffusion-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.

Anisotropic polygonal and polyhedral discretizations in finite element analysis

“…Since v ∈ H 1 (K) locally and due to the continuity of v across edges and faces for v ∈ V h , the conformity V h ⊂ H 1 (Ω) follows. A further adaptation of the approximation space in three-dimensions can be found in [14]. In order to achieve good approximation properties in V h the polytopal mesh and the elements in particular have to fulfil certain regularity assumptions.…”

Gradient Recovery for the BEM‐based FEM and VEM

“…Despite its newness, it has already been applied to a wide range of problems [12][13][14]. Other fields of application of the BEM-based FEM include, but are not limited to, convection dominated problems [22], anisotropic discretisations [23] and Nyström-based formulations [24]. The latter has been introduced in [18] and has been studied, in particular, for adaptive FEM strategies involving residual [19,20] and goal-oriented error estimators [21].…”

Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM

“…These problems are treated by means of Boundary Element Methods (BEM) that gave the name. The BEM-based FEM has been generalized to high order approximations [25,32,34], mixed formulations with H(div) conforming approximations [14] as well as to convection-adapted trial functions [18]. Furthermore, the strategy has been applied to general polyhedral meshes [26] and time dependent problems [33].…”