Journal of Scientific Computing

, Volume 76, Issue 1, pp 89–119 | Cite as

HiMod Reduction of Advection–Diffusion–Reaction Problems with General Boundary Conditions

  • Matteo C. Aletti
  • Simona Perotto
  • Alessandro Veneziani


We extend the hierarchical model reduction procedure previously introduced in Ern et al. (in: Kunisch, Of, Steinbach (eds) Numerical mathematics and advanced applications, Springer, Berlin, pp 703–710, 2008) and Perotto et al. (Multiscale Model Simul 8(4):1102–1127, 2010) to deal with general boundary conditions, enforcing their prescription in the basis function set. This is achieved by solving a Sturm–Liouville Eigenvalue problem. We analyze this approach and provide a convergence analysis for the associated error in the case of a linear advection–diffusion–reaction problem in rectangles (2D) and slabs (3D). Numerical results confirm the theoretical investigation and the reliability of the proposed approach.


Model reduction Spectral/finite element combined approximation Robin boundary conditions Sturm–Liouville Eigenvalue problem 

Mathematics Subject Classification

65N30 65N35 76M10 76M22 78M34 



The second and the third authors acknowledge the support of NSF Grant DMS-1419060 “Hierarchical Model Reduction Techniques for Incompressible Fluid-Dynamics and Fluid-Structure Interaction Problems” for this research. The authors wish to thank also Prof. Sandro Salsa for fruitful discussions concerning the analysis of the method, and Pablo J. Blanco for hosting the first author for one month at Laboratório Nacional de Computacao Científica in Petrópolis, Brazil. This article has been partially supported by Istituto Nazionale di Alta Matematica “Francesco Severi” (IT) Project (GNCS2017 Metodi numerici avanzati combinati con tecniche di riduzione computazionale per PDEs parametrizzate ed applicazioni).


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Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsInria Paris, Sorbonne Universités UPMC Univ Paris 6ParisFrance
  2. 2.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  3. 3.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  4. 4.School of Advanced Studies IUSSPaviaItaly

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