Title: The Discontinuous Enrichment Method (DEM) for Multi-Scale Transport Problems

Speaker: Irina Kalashnikova, Stanford University

Date/Time: Monday, February 7, 2011, 10:00 am in NM & 9:00 am in CA      

Location: CSRI Building/Room 90 (Sandia NM) and 915/W133 in SNL/CA

Brief Abstract: The discontinuous enrichment method (DEM) is among a number of different finite element approaches that have been proposed for addressing the challenge of solving the advection-diffusion equation

- div(k(x) grad c(x)) + a(x) . grad c(x) = f(x),

accurately and efficiently in the advection dominated (high Peclet) regime. This equation arises in its vector form in the linearization of the Navier-Stokes equations and is among the set of equations for which the standard finite element method (FEM) can be inadequate. The basic idea of DEM is to construct a finite element basis that is related to the operator governing the problem being solved, and therefore has a natural potential for resolving difficult features in the problem’s solution, such as sharp gradients. To this effect, the approximation space in DEM is defined as the set of free-space solutions of the homogeneous form of the governing PDE, obtained in analytical form. For the constant-coefficient advection-diffusion equation, the free-space solutions comprising the enrichment field are exponential functions that exhibit a steep gradient in the advection direction. These functions can be parametrized nicely with respect to a flow direction parameter so as to make possible the systematic design and implementation of DEM elements of arbitrary orders. The original constant-coefficient DEM methodology has a natural extension to variable-coefficient transport problems. In the variable-coefficient scenario, the approximation can be improved further by augmenting the enrichment space with additional free-space solutions to variants of the governing PDE. Since the enrichment in DEM is not constrained to vanish at the element boundaries, continuity of the solution across element interfaces in DEM is not automatic; rather, it is enforced weakly using Lagrange multipliers. The construction of several low and higher-order DEM elements fitting this paradigm is described. Numerical results on constant as well as variable-coefficient benchmark problems reveal that these DEM elements outperform their standard Galerkin and stabilized Galerkin counterparts of comparable computational complexity by a large margin, especially when the flow is advection-dominated. Some possible extensions of the method to non-linear and unsteady problems are discussed.

CSRI POC: Denis Ridzal, 505-845-1395