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. 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)
  2. 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)
  3. 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)
  4. Ganis, Benjamin; Juntunen, Mika; Pencheva, Gergina; Wheeler, Mary F.; Yotov, Ivan: A global Jacobian method for mortar discretizations of nonlinear porous media flows (2014)
  5. 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)
  6. Girault, Vivette; Vassilev, Danail; Yotov, Ivan: Mortar multiscale finite element methods for Stokes-Darcy flows (2014)
  7. Arbogast, Todd; Xiao, Hailong: A multiscale mortar mixed space based on homogenization for heterogeneous elliptic problems (2013)
  8. Pencheva, Gergina V.; Vohralík, Martin; Wheeler, Mary F.; Wildey, Tim: Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling (2013)
  9. Qu, Dong; Xu, Chuanju: Generalized polynomial chaos decomposition and spectral methods for the stochastic Stokes equations (2013)
  10. Tavener, Simon; Wildey, Tim: Adjoint based a posteriori analysis of multiscale mortar discretizations with multinumerics (2013)
  11. Ganis, Benjamin; Pencheva, Gergina; Wheeler, Mary F.; Wildey, Tim; Yotov, Ivan: A frozen Jacobian multiscale mortar preconditioner for nonlinear interface operators (2012)
  12. Wheeler, Mary Fanett; Xue, Guangri; Yotov, Ivan: A multiscale mortar multipoint flux mixed finite element method (2012)
  13. Cichosz, T.; Bischoff, M.: Consistent treatment of boundaries with mortar contact formulations using dual Lagrange multipliers (2011)
  14. Ganis, Benjamin; Yotov, Ivan; Zhong, Ming: A stochastic mortar mixed finite element method for flow in porous media with multiple rock types (2011)
  15. Wheeler, Mary F.; Wildey, Tim; Yotov, Ivan: A multiscale preconditioner for stochastic mortar mixed finite elements (2011)
  16. Wheeler, Mary F.; Wildey, Tim; Xue, Guangri: Efficient algorithms for multiscale modeling in porous media. (2010)
  17. Ganis, Benjamin; Yotov, Ivan: Implementation of a mortar mixed finite element method using a multiscale flux basis (2009)
  18. Girault, Vivette; Pencheva, Gergina V.; Wheeler, Mary F.; Wildey, Tim M.: Domain decomposition for linear elasticity with DG jumps and mortars (2009)