Implementation of a mortar mixed finite element method using a Multiscale Flux Basis. This paper provides a new implementation of a multiscale mortar mixed finite element method for second order elliptic problems. The algorithm uses non-overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem, which is then solved using an iterative method. The original implementation by T. Arbogast et al. [Simul. 6, No. 1, 319–346 (2007; Zbl 05255539)]. Multiscale model required solving one local Dirichlet problem on each subdomain per interface iteration. We alter this implementation by forming a Multiscale Flux Basis. This basis consists of mortar functions representing the individual flux responses for each mortar degree of freedom, on each subdomain independently. The computation of these basis functions requires solving a fixed number of Dirichlet subdomain problems. Taking linear combinations of the Multiscale Flux Basis functions replaces the need to solve any Dirichlet subdomain problems during the interface iteration. This new implementation yields the same solution as the original implementation, and is computationally more efficient in cases where the number of interface iterations is greater than the number of mortar degrees of freedom per subdomain. The gain in computational efficiency increases with the number of subdomains.

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  1. Arshad, Muhammad; Park, Eun-Jae: Multiscale mortar expanded mixed discretization of nonlinear elliptic problems (2020)
  2. Ahmed, Elyes; Fumagalli, Alessio; Budiša, Ana: A multiscale flux basis for mortar mixed discretizations of reduced Darcy-Forchheimer fracture models (2019)
  3. Ganis, Benjamin; Pencheva, Gergina; Wheeler, Mary F.: Adaptive mesh refinement with an enhanced velocity mixed finite element method on semi-structured grids using a fully coupled solver (2019)
  4. Khattatov, Eldar; Yotov, Ivan: Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry (2019)
  5. Ambartsumyan, Ilona; Khattatov, Eldar; Yotov, Ivan; Zunino, Paolo: A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model (2018)
  6. Arshad, Muhammad; Park, Eun-Jae; Shin, Dong-wook: Analysis of multiscale mortar mixed approximation of nonlinear elliptic equations (2018)
  7. Guiraldello, Rafael T.; Ausas, Roberto F.; Sousa, Fabricio S.; Pereira, Felipe; Buscaglia, Gustavo C.: The multiscale Robin coupled method for flows in porous media (2018)
  8. Kikinzon, Evgeny; Shashkov, Mikhail; Garimella, Rao: Establishing mesh topology in multi-material cells: enabling technology for robust and accurate multi-material simulations (2018)
  9. Ganis, Benjamin; Vassilev, Danail; Wang, ChangQing; Yotov, Ivan: A multiscale flux basis for mortar mixed discretizations of Stokes-Darcy flows (2017)
  10. Kikinzon, Evgeny; Kuznetsov, Yuri; Lipnikov, Konstatin; Shashkov, Mikhail: Approximate static condensation algorithm for solving multi-material diffusion problems on meshes non-aligned with material interfaces (2017)
  11. Arbogast, T.; Estep, D.; Sheehan, B.; Tavener, S.: A posteriori error estimates for mixed finite element and finite volume methods for parabolic problems coupled through a boundary (2015)
  12. Ganis, Benjamin; Juntunen, Mika; Pencheva, Gergina; Wheeler, Mary F.; Yotov, Ivan: A global Jacobian method for mortar discretizations of nonlinear porous media flows (2014)
  13. Ganis, Benjamin; Kumar, Kundan; Pencheva, Gergina; Wheeler, Mary F.; Yotov, Ivan: A global Jacobian method for mortar discretizations of a fully implicit two-phase flow model (2014)
  14. Girault, Vivette; Vassilev, Danail; Yotov, Ivan: Mortar multiscale finite element methods for Stokes-Darcy flows (2014)
  15. Arbogast, Todd; Xiao, Hailong: A multiscale mortar mixed space based on homogenization for heterogeneous elliptic problems (2013)
  16. Pencheva, Gergina V.; Vohralík, Martin; Wheeler, Mary F.; Wildey, Tim: Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling (2013)
  17. Qu, Dong; Xu, Chuanju: Generalized polynomial chaos decomposition and spectral methods for the stochastic Stokes equations (2013)
  18. Tavener, Simon; Wildey, Tim: Adjoint based a posteriori analysis of multiscale mortar discretizations with multinumerics (2013)
  19. Ganis, Benjamin; Pencheva, Gergina; Wheeler, Mary F.; Wildey, Tim; Yotov, Ivan: A frozen Jacobian multiscale mortar preconditioner for nonlinear interface operators (2012)
  20. Wheeler, Mary Fanett; Xue, Guangri; Yotov, Ivan: A multiscale mortar multipoint flux mixed finite element method (2012)

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