Towards a p-robust convergence analysis for Isogeometric Tearing and Interconnecting (IETI) methods

Dr. Stefan Takacs

Nov. 26, 2019, 3:30 p.m. S2 054

Isogeometric Analysis (IgA), as originally proposed, is a spline-based
finite element approach with global geometry function. Since more
complicated domains cannot be represented by just one such geometry
function, the whole domain is usually decomposed into subdomains, in IgA
typically called patches, where each of them is represented by its own
geometry function.

A standard Galerkin discretization yields a large-scale linear system.
For its solution, domain decomposition approaches are the methods of
choice. We consider a variant of the FETI-method with a standard scaled
Dirichlet preconditioner known as the Isogeometric Analysis Tearing and
Interconnecting (IETI) method. We will discuss convergence analysis,
focusing on the dependence of the condition number of the preconditioned
system on the spline degree p.

Previously, a convergence theory has been provided that is accurate
concerning the dependence of the condition number on the grid size.
There, an auxiliary problem with p=1 was introduced. The proof uses the
fact that the stiffness matrices of the original and the auxiliary
problem are spectrally equivalent. The constants in this spectral
equivalence are independent of the grid size but grow exponentially in p.

In this presentation, we will see a direct convergence proof, i.e., a
convergence proof that is solely based on the spline spaces of interest
and that does not use such an auxiliary problem. This allows us to get
rid of the exponential dependence in p. We will discuss the dependence
of all estimates on p.