Efficient solution procedures for adaptive finite element methods: Applications to elliptic problems

The goal of this thesis is to develop methods that will improve the computational efficiency of simulations performed by adaptive finite element methods (FEM). The two main areas that will be studied are efficient dynamic mesh partitioning strategies to divide the work efficiently among different processors and parallel solution schemes that decrease the solution time. In FEM simulations, the time needed to get a solution uses $O$(Nα) units of time where N is the number of degrees of freedom (DOF) of the mesh and α is a number usually between two and three depending on the efficiency of the solution process. The solution accuracy is also dependent on the mesh size. Thus, the higher the desired accuracy of the simulation, the longer it will take to compute the approximate solution. If the mesh is ‘optimized’ to the specific problem, the solution accuracy can be improved upon without significantly increasing the DOF of the mesh. While finding a ‘good’ mesh is a good start to decreasing the solution time, more can be done by using distributed memory parallel computers instead of stand-alone computers. There are difficulties inherent in creating programs that run efficiently on parallel machines that are not an issue in creating programs that run on stand-alone computers. One of these difficulties is assigning work properly to each processor on parallel machines. To address this difficulty we will develop and analyze a new Space Filling Curve (SFC) geometric partitioning algorithm. Other difficulties arise from the complexities of data management in this environment. We will discuss here schemes to manage these complexities in a simple and efficient manner. After a good mesh has been created and the work has been properly assigned to all of the processors, the solution of linear systems of equations arising from the discretization must be calculated. Traditionally this has required the use of iterative solution schemes based on the Krylov space methodology. However, these schemes usually degrade dramatically when adaptive methodologies are used. We will develop here good techniques to precondition the linear systems for efficient solution schemes even in the presence of adaptivity. We will demonstrate that the techniques developed here will reduce the solution cost greatly.