Research Spotlights

When the Research Spotlights section was launched in 2012, its scope was defined to include software package descriptions. This issue contains the first paper of this type to appear in the section. It describes the software package IFISS, for incompressible flow problems. The paper highlights the usefulness of IFISS as a toolbox for exploring the numerical properties of various formulations and algorithms, using problems with different properties. Users can observe, for example, that the use of higher-order finite elements does not improve the order of convergence on problems with singular solutions. They can also experiment with different types of preconditioning for linear systems. The design of the toolbox encourages users to experiment with different combinations of discretization schemes and iterative linear system solvers, giving a useful starting point for student projects as well as more advanced research activity. The IFISS package traces its origins in the mid-1990s and has been extended since, in many ways. It can be run from MATLAB or its public-domain counterpart Octave. The second paper in Research Spotlights examines the Helmholtz equation, a classical system that can arise when separation of variables is applied to such PDEs as the wave equation. A property of this equation, well known to numerical analysts, is that standard weak (variational) formulations are sign-indefinite for interesting values of the wave number. That is, the coefficient matrix in their discretization is indefinite. The paper shows that for certain kinds of boundary conditions and domain shapes, it is possible to derive a sign-definite formulation. The usual variational framework is used, but there are clever choices of the test functions and of the identity to be integrated (by parts). This sign-definiteness property potentially broadens the class of linear algebra techniques that can be applied to the discretized problem. The authors make a start on analyzing conditioning properties and convergence of the discretized system. They conclude with a wish that future statements about the sign-indefiniteness of the Helmholtz equation will at least be qualified with an asterisk!