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Advanced Numerical Methods in the Mathematical Sciences

Victor Calo, King Abdullah University of Science and Technology
PetIGA: High-Performance Isogeometric Analysis


  • V.M. Calo
  • N.O. Collier
  • A.M.A. Cortes
  • L.A. Dalcin
  • A. Sarmiento
  • P. Vignal


We have developed fast implementations of B-spline/NURBS based finite element analysis, written using PETSc. PETSc is frequently used in software packages to leverage its optimized and parallel implementation of solvers; however, we also are using PETSc data structures to assemble the linear systems. These structures were originally intended for the parallel assembly of linear systems resulting from finite differences. We have reworked this structure for linear systems resulting from isogeometric analysis based on tensor product spline spaces. The result of which is the PetIGA framework for solving problems using isogeometric analysis which is scalable and greatly simplified over previous solvers.

Our infrastructure has also allowed us to develop scalable solvers for a variety of problems. We have chosen to pursue nonlinear time dependent problems, such as:

  • Cahn-Hilliard
  • Navier-Stokes-Korteweg
  • Variational multiscale for Navier-Stokes
  • Diffusive wave approximation to shallow water equations
  • Phase-field crystal equation and its time integration
  • Divergence-conforming B-spline model for nanoparticle suspensions

We also have solvers for an assortment of linear problems: Poisson, elasticity, Helmholtz, thin shells, advection-diffusion, and diffusion-reaction. All solvers are written to be inherently parallel and run on anything from a laptop to a supercomputer such as Shaheen, KAUST's IBM-BlueGeneP supercomputer. In this presentation, we will focus on new time integration techniques for phase-field modeling which are energy stable and allow for stable linearizations of the underlying non-linear model as well as on divergence conforming discretizations for nanoparticle suspensions.