Stefan Takacs

FWF Stand-alone project P31048

Title Fast Solvers for Isogeometric Analysis
Funding agency FWF Austrian Science Fund (FWF)
Project number P31048
Principal investigator Stefan Takacs
Approval date 19 Mar 2018
Project employee Jarle Sogn

Isogeometric Analysis (IgA) is a paradigm aiming to improve the connection between the world of computer aided design (CAD) and the world of finite element (FEM) simulation by using spline functions or other functions commonly used in CAD also for the FEM simulation. The use of high-order functions in IgA yields approximation errors that are smaller than those of standard FEM. Unlike in standard high-order FEM approaches, which show similar approximation errors, in IgA increasing the spline degree basically does not increase the number of unknowns. This does not come for free: the condition numbers of the resulting mass and stiffness matrices grow exponentially with the spline degree.

The main research interest of the principal investigator is the construction and the analysis of fast numerical methods for solving linear systems arising from the discretization of partial differential equations with isogeometric functions. It is typically possible to carry over ideas from FEM to IgA with minimal adaptions. However, often one obtains methods that are rather inefficient, particularly if the spline degree is increased. Starting form simple problems (particularly the Poisson problem), it has been shown that the steps of numerical simulation, like assembling of the matrices or solving the resulting linear systems, can be done in a much better way if one uses particular features of the isogeometric function spaces. One contribution to this area was archived by the principal investigator and his coworkers by developing a new multigrid solver, whose convergence rates are provable robust in the spline degree and the grid size.

The main goal of the proposed project is to go beyond that multigrid solver in two directions: First, more complicated partial differential equations, like the Stokes equation should be considered. Due to its saddle point structure, certain additional challenges arise, which should be faced in the project. Stable discretizations and solvers are available but the dependence of the approximation errors and the convergence rates on the spline degree is not analyzed. In the project, the analysis should be carried out using classical approaches including Brezzi's conditions or tools like Fourier analysis.

Second, more complicated computational domains should be considered. So far, the above-mentioned multigrid solver works for the single-patch case, i.e., domains which are described with one global geometry function. More complicated domains have to be decomposed into patches with their own geometric description. For these problems, ideas from domain decomposition seem to be of particular interest. Such ideas could be used for setting up a multigrid smoother for the whole multi-patch domain or, e.g., in the framework of the IETI-method, a FETI-like approach for IgA.