Reference documentation for deal.II version 8.4.1
deal.II's tutorial contains a set of programs that together form the geodynamics demonstration suite. The idea of these programs is to demonstrate techniques for advanced finite element software using applications from geodynamics, i.e. the investigation of processes in the solid earth. By doing so, these programs are supposed to provide a basis for more specialized, dedicated programs that can solve actual geodynamics problems, for example as part of the work of graduate students or postdocs. A more thorough discussion of the motivation for these programs follows below.
Currently, the geodynamics testsuite contains the following programs:
Some of these programs were developed under contract from the California Institute of Technology with support by the National Science Foundation under Award No. EAR-0426271, the grant that funded the Computational Infrastructure in Geodynamics initiative. The recipient, Wolfgang Bangerth, gratefully acknowledges this source of support.
Adaptive mesh refinement (AMR) has long been identified as a key technology that would aid in the accurate and efficient numerical solution of a number of geodynamics applications. It has been discussed in the geodynamics community for several years and has been a continuous topic on the task list of CIG since its inception. Yet, relatively little has happened in this direction so far. Only recently have there been attempts to use AMR in geodynamics: CIG sponsored a workshop on AMR technique in Boulder in October 2007, and a collaboration between George Biros, Omar Ghattas, Mike Gurnis, and Shijie Zhong's groups is currently developing a parallel adaptive mantle convection solver.
One of the reasons for the slow adoption of AMR techniques in geodynamics is the relatively steep initial hurdle: codes have to provide the data structures and algorithms to deal with adaptive meshes, finite elements have to be able to deal with hanging nodes, etc. To do so efficiently and in sufficient generality adds several 10,000 lines of code to finite element programs, too much for the average student to do within the time frame of a dissertation. On the other hand, there are libraries that provide the infrastructure code on which applications supporting AMR can rapidly be built. deal.II of course provides exactly this infrastructure.
The goal of the geodynamics testsuite is to write programs for a variety of topics relevant to geodynamics. Continuing in the style of the existing tutorial programs – an extensive introduction explaining the background and formulation of an application as well as the concepts of the numerical scheme used in its solution; detailed comments throughout the code explaining implementation details; and a section showing numerical results – we intend to provide the resulting programs as well-documented applications solving model problems. In particular, they are aimed at the following goals:
Starting points: The existing tutorial of deal.II has proven to be an excellent starting point for graduate students and researchers to jump-start developing their own applications. By providing programs that are already close to the targeted application, first results can often be obtained very quickly, both maintaining the initial enthusiasm during development as well as allowing to spend research time on implementing application specific behavior rather than using months of work on basic infrastructure code supporting AMR.
Supporting this point is the fact that although there are currently at least 170 publications presenting results obtained with deal.II, we are aware of only a handful of applications that have been built with deal.II from scratch; all others have started as modifications of one of the tutorial programs.
Training: The tutorial programs we propose to write will provide students and researchers with a reference implementation of current numerical technology such as AMR, higher order elements, sophisticated linear and nonlinear solvers, stabilization techniques, etc. Providing these as starting points for further development by others will also serve the goal of training a new generation of geodynamicists in modern numerical algorithms.
Extending equations and formulations: In deal.II, it is fairly simple to extend a set of equations by another equation, for example an additional advected quantity that enters the existing equations as a right hand side or in one of the coefficients. Since applications typically use blocked matrices rather than the one-big-matrix-for-everything approach, it is also not complicated to find suitable linear solvers for augmented equations. Consequently, deal.II is a good tool for trying out more complex formulations of problems, or more complete models and their effects on the accuracy of solutions.
Rapid prototyping and benchmarking: deal.II provides many interchangeable components that allow rapid prototyping of finite element kinds and orders, stabilization techniques, or linear solvers. For example, typically only a few lines of code have to be changed to replace low-order by high-order elements. Through this, it becomes relatively simple to try out higher order elements, a different block elimination solver, or a different stabilization technique. In turn, this may help in benchmarking applications both regarding computing times to solve as well as concerning the accuracy of numerical solutions.
The applications in this module will already have been benchmarked for correctness. Existing tutorial programs typically employ simpler rather than more complicated solver schemes for exposition but frequently suggest more complicated schemes including hints on how they might be implemented in an appendix.