Isogeometric Residual Minimization Method (iGRM) with direction splitting for non-stationary advection–diffusion problems

M. Łoś, J. Muñoz-Matute, I. Muga, M. Paszyński

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product B-spline basis functions in space, implicit second order time integration schemes, residual minimization in every time step, and we exploit Kronecker product structure of the matrix to employ linear computational cost alternating direction solver. We implement an implicit time integration scheme and apply, for each space-direction, a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method on three advection–diffusion computational examples, including model “membrane” problem, the circular wind problem, and the simulations modeling pollution propagating from a chimney.

Original languageEnglish
Pages (from-to)213-229
Number of pages17
JournalComputers and Mathematics with Applications
Issue number2
StatePublished - 15 Jan 2020


  • Advection-diffusion simulations
  • Implicit dynamics
  • Isogeometric analysis
  • Linear computational cost
  • Residual minimization
  • Second order time integration schemes


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