A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes

Abstract

SUMMARYA higher-order discontinuous enrichment method (DEM) with Lagrange multipliers is proposed for the efficient finite element solution on unstructured meshes of the advection-diffusion equation in the high Péclet number regime. Following the basic DEM methodology, the usual Galerkin polynomial approximation is enriched with free-space solutions of the governing homogeneous partial differential equation (PDE). In this case, these are exponential functions that exhibit a steep gradient in a specific flow direction. Exponential Lagrange multipliers are introduced at the element interfaces to weakly enforce the continuity of the solution. The construction of several higher-order DEM elements fitting this paradigm is discussed in details. Numerical tests performed for several two-dimensional benchmark problems demonstrate their computational superiority over stabilized Galerkin counterparts, especially for high Péclet numbers.

A discontinuous enrichment method for variable‐coefficient advection–diffusion at high Péclet number

“…The functions are identical to those derived and parametrized in [15,16]. Here, the point x e r,i ≡ (x e r,i , y e r,i ) is an arbitrary reference point for the ith enrichment function c E (x; i ), introduced within 316 I. KALASHNIKOVA, R. TEZAUR AND C. FARHAT Figure 2.…”

“…It is straightforward to show [15,16] that on a regular mesh of n el quadrilateral elements, this condition implies the asymptotic bound on the number of Lagrange multipliers per edge (n ) given by…”

“…In practice, fewer than n = n E /2 Lagrange multipliers per edge are employed. Numerical tests [15,16] show that the general rule of thumb is to limit…”

“…To this effect and in order to keep this paper as self-contained as possible, the enrichment field for the case of the constant-coefficient advection-diffusion equation (1) is first overviewed. For further details on this topic, the reader is referred to [15,16].…”

“…In variational multiscale (VMS) terminology, the splitting of the approximation into polynomials and enrichment functions, as done in the first line of (15), can be viewed as a decomposition of the numerical solution into coarse (polynomial) and fine (enrichment) scales. Elements for which the solution space V h is constructed as a direct sum of V P and V E are termed genuine or 'full' DEM elements.…”