A multiscale hybrid method

Gabriel R. Barrenechea, Antonio Tadeu A. Gomes, Diego Paredes

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In this work we propose, analyze, and test a new multiscale finite element method called Multiscale Hybrid (MH) method. The method is built as a close relative to the Multiscale Hybrid Mixed (MHM) method, but with the fundamental difference that a novel definition of the Lagrange multiplier is introduced. The practical implication of this is that both the local problems to compute the basis functions, as well as the global problem, are elliptic, as opposed to the MHM method (and also other previous methods) where a mixed global problem is solved, and constrained local problems are solved to compute the local basis functions. The error analysis of the method is based on a hybrid formulation, and a static condensation process is done at the discrete level, so the final global system only involves the Lagrange multipliers. We tested the performance of the method by means of numerical experiments for problems with multiscale coefficients, and we carried out comparisons with the MHM method in terms of performance, accuracy, and memory requirements.
Original languageEnglish
Pages (from-to)A1628 - A1657
Number of pages30
JournalSIAM Journal on Scientific Computing
Issue number3
Early online date13 May 2024
Publication statusE-pub ahead of print - 13 May 2024


  • multiscale finite element;
  • hybrid problem
  • multiscale diffusion equation.


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