Nathan Collier

Nathan Collier, Lisandro Dalcin, David Pardo et al.

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using $C^0$ B-splines, which span traditional finite element spaces, and $C^{p-1}$ B-splines, which represent maximum continuity. We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size $h$ and polynomial order of approximation $p$. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most $\slfrac{33p^2}{8}$ times more expensive for the more continuous space, although for moderately low $p$, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high $p$. Preconditioning options can be up to $p^3$ times more expensive to setup, although this difference significantly decreases for some popular preconditioners such as Incomplete LU factorization.

Victor M. Calo, Nathan Collier, Matthias Gehre et al.

We study the numerical recovery of Manning's roughness coefficient for the diffusive wave approximation of the shallow water equation. We describe a conjugate gradient method for the numerical inversion. Numerical results for one-dimensional model are presented to illustrate the feasibility of the approach. Also we provide a proof of the differentiability of the weak form with respect to the coefficient as well as the continuity and boundedness of the linearized operator under reasonable assumptions using the maximal parabolic regularity theory.

Nathan Collier, David Pardo, Maciej Paszynski et al.

The multi-frontal direct solver is the state-of-the-art algorithm for the direct solution of sparse linear systems. This paper provides computational complexity and memory usage estimates for the application of the multi-frontal direct solver algorithm on linear systems resulting from B-spline-based isogeometric finite elements, where the mesh is a structured grid. Specifically we provide the estimates for systems resulting from $C^{p-1}$ polynomial B-spline spaces and compare them to those obtained using $C^0$ spaces.

Lisandro Dalcin, Nathan Collier, Philippe Vignal et al.

We present PetIGA, a code framework to approximate the solution of partial differential equations using isogeometric analysis. PetIGA can be used to assemble matrices and vectors which come from a Galerkin weak form, discretized with Non-Uniform Rational B-spline basis functions. We base our framework on PETSc, a high-performance library for the scalable solution of partial differential equations, which simplifies the development of large-scale scientific codes, provides a rich environment for prototyping, and separates parallelism from algorithm choice. We describe the implementation of PetIGA, and exemplify its use by solving a model nonlinear problem. To illustrate the robustness and flexibility of PetIGA, we solve some challenging nonlinear partial differential equations that include problems in both solid and fluid mechanics. We show strong scaling results on up to 4096 cores, which confirm the suitability of PetIGA for large scale simulations.