Fast Solvers For Isogeometric Analysis

Isogeometric Analysis was proposed in the year 2005 as a new method to calculate approximate solutions of partial differential equations. It can be seen as an extension or a variant of the finite element method. Partial differential equations arise in many areas, including science and engineering. Examples for partial differential equations are the Poisson equation describing the heat transfer or the Stokes equation describing the flow of a liquid or a gas. Isogeometric Analysis was originally proposed to allow solving partial differential equations directly on the geometric objects from computer aided design (CAD) software. So, using this technique one avoids one step that is done in standard finite element approaches: approximating the geometric object by a mesh of triangles or tetrahedrons. Moreover, in Isogeometric Analysis spline functions are used to approximate the solution and those functions can approximate the solution better than the piecewise linear functions from standard finite elements. The main research interest of the proposer is focused onto efficient algorithms for solving the linear systems arising from isogeometric discretizations. If one uses solvers that have been designed to work for standard finite element discretizations in Isogeometric Analysis only with minimal adaptions, one typically obtains methods that work, but which are less efficient than in the finite element case. If the spline degree is increased, these methods become even more inefficient, which sweeps off the advantages of Isogeometric Analysis. The proposer and his coworkers have shown that it is possible to modify the well-known multigrid method such that it works well for any spline degree. So far, this method has been worked out only for the Poisson equation which is commonly known as the simplest partial differential equation. Of course, it is of interest to be able to solve also other, more challenging differential equations. In the project, exactly this should be done, with a focus onto the Stokes equations. First, simple geometric objects that can be represented by one global geometry function should be treated. Then, in a second step, the methods should be extended to cases where the geometrical object of interest is subdivided into subobjects (like a turbine can be subdivided into its blades) that are described by their own geometry function. For such domains, robust solvers working well for all spline degrees are still to be developed, both for the Poisson equation and the Stokes equations. With the new methods, it will be possible to solve the linear systems arising from the discretizations efficiently. This does not only allow to solve for the equations faster, but it also allows to consider more complex geometric domains or to archive more accurate solutions by stronger refinement.

Computer simulation can be used in order to determine stresses or deformations, to determine the flow of liquids or gases or to determine the heat distribution in some object of interest, like a component of a machine to name just an example. Such objects are often designed with CAD (Computer Aided Design) software; simulations are usually done with FEM (Finite Element Method) software. Typically, these two kinds of software tools use completely different methods to represent the object of interest. The goal of Isogeometric Analysis is to bridge the gap between CAD and FEM by establishing a unified way of representing the object. When it comes to the realization of that idea, there are a few issues that have to be addressed, since one typically wants the simulation to deliver accurate results, one wants to be able to estimate the error made by the approximation, one wants the result to depend on the input data in a stable way, one wants the algorithms used for simulation to use as few computation steps (and therefore as little time) and as little memory as possible. In the project, we have addressed exactly these issues. We have focused to the Poisson equation (that models for example the heat distribution), the Stokes equations (which model flows of liquids or gases) and the equations for linearized elasticity (which models the deformation of solid objects). These differential equations are discretized with Isogeometric Analysis, this means, we approximate them with linear systems of equations. By approximating the true solution with the solution to the discretized problem, a small error is made. We have proven that the error cannot exceed a certain upper bound that depends on the grid size and other factors, like the chosen polynomial degree. Moreover, we have proposed algorithms to solve the linear systems. We have tested these algorithms by implementing them on a computer. Additionally, we could mathematically prove that these algorithms cannot exceed certain upper bounds concerning the number of computation steps and concerning the required memory. These upper bounds depend on the specific differential equation, the shape of the object in question, and the discretization, specifically the grid size. If the grid is refined, the quality of the approximate solution is improved on the one hand, but the number of unknowns to be solved for is increased on the other hand. Unavoidably, this means that one needs more computation steps and memory, in the best case, this increase in computation steps and memory requirements is linear. For some of the proposed algorithms, we could prove that the increase is actually linear, for the other algorithms, we could prove upper bounds that grow only slightly faster than a linear increase would do.