The anaerobic biodegradation mechanisms of linear alcohol ethoxylates (LAE) were studied in incubation experiments with anoxic sewage sludge. Sophisticated analytical techniques were applied, such as solid-phase extraction (SPE) followed by reversed phase high performance liquid chromatography (HPLC) procedures based on the derivatization of LAE and poly(ethylene glycol) (PEG). During the degradation of LAE C 12 (EO) ˜9, a technical dodecanol ethoxylate with an average of nine ethoxy (EO) units, and LAE C 12 (EO) 8 , a single ethoxymer, alcohol ethoxylates with shortened EO chains were released as the first identifiable metabolites, but no PEG products were observed. From our results it was concluded that the first step of anaerobic microbial attack on the LAE molecule is the cleavage of the terminal EO unit, releasing acetaldehyde stepwise, and shortening the ethoxy chain until the lipophilic moiety is reached. In contrast to the aerobic degradation pathway, where central scission prevails (the cleavage of the ether bond between alkyl and ethoxy chains), such a primary attack on the surfactant molecule is very unlikely in an anaerobic community of fermenting bacteria.

We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the solution at the scale of interest at computational cost independent of the small scale by performing numerical upscaling (coupling of macro and micro finite element methods). Optimal a priori error estimates in the L 2 (H 1 ) and C 0 (L 2 ) norm are derived taking into account the error due to time discretization as well as macro and micro spatial discretizations. Further, we present numerical simulations to illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.

We introduce and analyze an efficient numerical homogenization method for a class of nonlinear parabolic problems of monotone type in highly oscillatory media. The new scheme avoids costly Newton iterations and is linear at both the macroscopic and the microscopic scales. It can be interpreted as a linearized version of a standard nonlinear homogenization method. We prove the stability of the method and derive optimal a priori error estimates which are fully discrete in time and space. Numerical experiments confirm the error bounds and illustrate the efficiency of the method for various nonlinear problems.Keywords: monotone parabolic multiscale problem, linearized scheme, numerical homogenization method, fully discrete a priori error estimates.

We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the homogenized solution at computational cost independent of the small scale by performing numerical upscaling (coupling of macro and micro finite element methods). Taking into account the error due to time discretization as well as macro and micro spatial discretizations, the convergence of the method is proved in the general L p (W 1,p ) setting. For p = 2, optimal convergence rates in the L 2 (H 1 ) and C 0 (L 2 ) norm are derived. Numerical experiments illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.

A discontinuous Galerkin finite element heterogeneous multiscale method is proposed for advectiondiffusion problems with highly oscillatory coefficients. The method is based on a coupling of a discontinuous Galerkin discretization for an effective advection-diffusion problem on a macroscopic mesh, whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element. The computational work involved is independent of the high oscillations in the problem at the smallest scale. The stability of our method (depending on both macro and micro mesh sizes) is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data. Fully discrete a priori error bounds are derived for locally periodic data. Numerical experiments confirm the theoretical error estimates.