A priori error estimate of a multiscale finite element method for transport modeling

Abstract

This work proposes an a priori error estimate of a multiscale finite element method to solve convection-diffusion problems where both velocity and diffusion coefficient exhibit strong variations at a scale which is much smaller than the domain of resolution. In that case, classical discretization methods, used at the scale of the heterogeneities, turn out to be too costly. Our method, introduced in [3], aims at solving this kind of problems on coarser grids with respect to the size of the heterogeneities by means of particular basis functions. These basis functions are defined using cell problems and are designed to reproduce the variations of the solution on an underlying fine grid. Since all cell problems are independent from each other, these problems can be solved in parallel, which makes the method very efficient when used on parallel architectures. This article focuses on the proof of an a priori error estimate of this method.

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

“…For parabolic advection-diffusion PDEs with Péclet number of order O(ε −1 ) but divergence free advection, multiscale methods have been proposed in [1,33,47]. In [1] a method to solve advection-diffusion problems with constant scalar diffusion is introduced, where the advection is obtained as the flux of an elliptic diffusion multiscale equation which is solved using a finite element heterogeneous multiscale method (FE-HMM), see [28] and [2,3] for general reviews of heterogeneous multiscale methods (HMM) and FE-HMM, respectively, and stabilized explicit Runge-Kutta methods (ROCK) [8] serve as time integrator.…”

“…Further, a posteriori error control is discussed in [32]. In [48] a space-discrete multiscale method extending the multiscale finite element method (MsFEM) from [12] has been proposed (see [35] for an MsFEM applied to cellular flow). The usual finite element basis functions are replaced by solutions to elliptic advection-diffusion problems on patches of macroscopic size with Dirichlet boundary conditions.…”

“…As the Péclet number within the localized problems is still large, the basis functions in general have boundary layers. For periodic data, an a priori error analysis with explicit rates for the convergence of the numerical solution towards the finescale solution is given in [47]. However, the convergence rates are not robust for ε → 0.…”

“…The proposed numerical upscaling procedure captures the effective dynamics for general flows b ε as neither div b ε = 0 nor zero mean for b ε is assumed, while the methods proposed in [33,47] are only designed for incompressible flows b ε . Our main a priori error estimates are also valid for periodic flows b ε without assuming div b ε = 0 and they are robust with respect to the Péclet number which scales as O(ε −1 ).…”

“…Our main a priori error estimates are also valid for periodic flows b ε without assuming div b ε = 0 and they are robust with respect to the Péclet number which scales as O(ε −1 ). In contrast, the estimates in [33,47] are only valid for incompressible periodic flows. Further, in [33] a priori error estimates are derived in a moving frame (i.e., an analysis of a pure diffusion problem) while the a priori error bounds in [47] are not robust for ε → 0 (as only the right-hand side of their estimates has terms scaling as O(ε −1 )).…”