[Paper Review] PetIGA-MF: a multi-field high-performance toolbox for structure-preserving B-splines spaces

Adel F. Sarmiento, Adriano Cortês+4 more|arXiv (Cornell University)|Feb 28, 2016

Advanced Numerical Analysis Techniques19 references4 citations

TL;DR

PetIGA-MF is a high-performance, multi-field isogeometric analysis framework that leverages structure-preserving B-spline spaces to solve complex fluid dynamics problems with optimal convergence and stability. It extends PetIGA with advanced discrete differential forms for gradient-, curl-, divergence-, and integral-conforming spaces, achieving optimal convergence rates across Darcy, Stokes, Brinkman, and Navier-Stokes benchmarks, even under mesh distortion.

ABSTRACT

We describe the development of a high-performance solution framework for isogeometric discrete differential forms based on B-splines: PetIGA-MF. Built on top of PetIGA, PetIGA-MF is a general multi-field discretization tool. To test the capabilities of our implementation, we solve different viscous flow problems such as Darcy, Stokes, Brinkman, and Navier-Stokes equations. Several convergence benchmarks based on manufactured solutions are presented assuring optimal convergence rates of the approximations, showing the accuracy and robustness of our solver.

Motivation & Objective

• To develop a high-performance, multi-field isogeometric framework that preserves the mathematical structure of partial differential equations in discrete spaces.
• To enable stable and accurate simulations of incompressible viscous flows using divergence- and integral-conforming B-spline discretizations.
• To extend PetIGA with support for multiple compatible finite element spaces, including curl-, gradient-, divergence-, and integral-conforming formulations.
• To evaluate the robustness and convergence behavior of the solver under mesh distortion and varying polynomial orders.
• To provide a scalable, parallelized implementation leveraging PETSc for efficient solution of large-scale problems.

Proposed method

• The framework uses isogeometric discrete differential forms based on B-spline basis functions to construct structure-preserving finite element spaces.
• It implements a weak Galerkin formulation for the generalized Navier-Stokes problem, incorporating mixed velocity-pressure pairs with compatible function spaces.
• Boundary conditions are weakly imposed via Nitsche’s method, preserving stability and accuracy near boundaries.
• The solver leverages PETSc’s parallel linear algebra and iterative solvers, enabling high-performance computation on distributed architectures.
• A priori error estimates and convergence benchmarks are used to validate optimal convergence rates across different polynomial orders and mesh types.
• The framework supports both uniform and distorted meshes, enabling analysis of geometric robustness.

Experimental results

Research questions

• RQ1 Can a high-performance isogeometric framework achieve optimal convergence rates for multi-field fluid problems using structure-preserving B-spline spaces?
• RQ2 How does mesh distortion affect the convergence behavior of velocity and pressure approximations in divergence-conforming discretizations?
• RQ3 To what extent do the discrete spaces guarantee point-wise divergence-free velocity fields and inf-sup stability?
• RQ4 Can weak enforcement of boundary conditions maintain accuracy while avoiding over-constraining the velocity space?
• RQ5 How does the choice of polynomial order and mesh refinement impact the accuracy of Darcy, Stokes, Brinkman, and Navier-Stokes solutions?

Key findings

• The framework achieves optimal convergence rates for both velocity and pressure fields across all test cases, including Darcy, Stokes, Brinkman, and Navier-Stokes problems.
• For uniform meshes, pressure convergence matches velocity convergence despite using one order lower polynomial spaces, indicating superconvergence.
• Under mesh distortion, pressure convergence drops by nearly one order of magnitude, approaching the theoretical limit predicted by a priori error estimates.
• The results suggest that the error estimates in [8] are not overly conservative, but optimal convergence may only be achieved under ideal mesh conditions.
• Weakly enforced boundary conditions yield accurate results near boundaries without inducing instabilities or over-constraining the velocity space.
• Numerical results for Navier-Stokes flow at Re=100 and Re=400 show excellent agreement with benchmark solutions, with results indistinguishable from Wong’s benchmark at Re=100.