Matthew Knepley

Robert C. Kirby, Matthew Knepley, Anders Logg et al.

Assembling stiffness matrices represents a significant cost in many finite element computations. We address the question of optimizing the evaluation of these matrices. By finding redundant computations, we are able to significantly reduce the cost of building local stiffness matrices for the Laplace operator and for the trilinear form for Navier-Stokes. For the Laplace operator in two space dimensions, we have developed a heuristic graph algorithm that searches for such redundancies and generates code for computing the local stiffness matrices. Up to cubics, we are able to build the stiffness matrix on any triangle in less than one multiply-add pair per entry. Up to sixth degree, we can do it in less than about two. Preliminary low-degree results for Poisson and Navier-Stokes operators in three dimensions are also promising.

Darsh Nathawani, Matthew Knepley

We present a one-dimensional shear-force-driven droplet formation model with a flux-based error estimator. The model is derived using asymptotic expansion and a front-tracking method to simulate the droplet interface. The model is then discretized using the Galerkin finite element method in the mixed form. However, the solution gradients exhibit large jumps across element boundaries and can grow rapidly due to the highly convective pinch-off process. This leads to an erroneous droplet interface and incorrect curvature. Therefore, the mesh must be sufficiently refined to capture the interface accurately. The mixed form of the governing equation naturally provides smooth interface gradients that can be used to compute the error estimate. The computed error estimate is then used to drive the adaptive mesh refinement algorithm. The efficacy of the error estimator is illustrated by comparing the droplet profiles obtained with adaptive refinement to those obtained with regular refinement. The adaptive mesh refinement approach reduces the computational cost significantly without compromising accuracy.

Mark Adams, Satish Balay, Oana Marin et al.

The communities who develop and support open source scientific software packages are crucial to the utility and success of such packages. Moreover, these communities form an important part of the human infrastructure that enables scientific progress. This paper discusses aspects of the PETSc (Portable Extensible Toolkit for Scientific Computation) community, its organization, and technical approaches that enable community members to help each other efficiently.