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We present a new discretization method for convection‐diffusion‐reaction boundary value problems in 3D 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 as 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. Our experiments also show an improved resolution of the exponential layer at the outflow boundary for our proposed method when compared to the SUPG method.

Convection‐adapted BEM‐based FEM