FOSPACK: A FOSLS/AMG Package

John Ruge

1005 Gillaspie Dr. Boulder, CO 80303


Abstract

First order system least squares (FOSLS) is a recent development by Manteuffel and McCormick for expressing and discretizing systems of PDEs. In this method, the original second-order system is reduced to a first-order system by introducing auxiliary unknowns, generally derivatives of the primary unknowns. Some "redundant" equations can also be introduced in order to make the final problem more well-posed. A FOSLS functional, consisting of the sum of the integrals of the residuals of the first-order equations, is then formed, so that the problem is recast as a minimization problem for this functional over the permissible spaces. Boundary conditions are either incorporated into the functional, or imposed on the spaces involved. Now, the discrete problem is then formed by choosing suitable finite element spaces for the unknowns, and setting up the minimization equations. This approach has a number of advantages, including a natural error measure (the functional value), and more freedom in choosing the finite element spaces than with traditional methods. In addition, when properly formulated, the equations for the different unknowns are somewhat decoupled, and the diagonal blocks (the operators coupling each unknown with itself) are nicely elliptic. Together, these say that a block- diagonal preconditioner would work well, and multigrid or multigrid-like methods sould be very effective for inversion of the diagonal blocks. This is supported by both theory and numerical tests. Algebraic multigrid (AMG) has been shown to work quite well on elliptic problems, and is particularly useful with unstructured meshes, complicated domains, and varying or discontinuous problem coefficients. In the development of test codes over the past few years, it has become obvious that combining FOSLS and AMG could be very useful, and FOSPACK (First-Order System PACKage) is the result. The intent is to allow a user to quickly specify a domain (and initial mesh) and the first-order system, and have the package do the rest. The current 2D package allows for fairly arbitrary (user-generated) domains/ meshes or more simple (FOSPACK-generated) domains. A flexible interpreter allows for a simple specification of the first-order system (which itself can be quite complicated, involving spatially varying coefficients, matrix/vector notation, and more) in a data file. Other features are automatic linearization of some nonlinear first-order problems (with iteration), and the use of "zones" to allow different types of equations to be specified in different subdomains. Current work includes development of a graphical user interface and local refinement capabilities, while future work includes more extensive mesh- generation options and extension to 3D. This talk will include a quick overview of FOSLS and AMG, a description of the different components of the code, and sample problem specifications and results.