# Condition number bounds for IETI-DP methods that are explicit inh and p

## Dr. Stefan Takacs

**Nov. 17, 2020, 3:30 p.m. ZOOM**

Isogeometric Analysis (IgA) is a spline-based nite elementapproach with global geometry function. Following the usualapproach, the overall computational domain is decomposed intosubdomains, in IgA typically called patches, where each of them isrepresented by its own geometry function. A standard Galerkindiscretization yields a large-scale linear system.

For the solution of that linear system, domain decompositionapproaches are a canonical choice since they can be based on thesubdivision of the overall domain into patches. We consider theIsogeometric Analysis Tearing and Interconnecting (IETI) method,which is a variant of the FETI-method for IgA. We will discussconvergence analysis, focusing on the dependence of the conditionnumber of the preconditioned system on the spline degree p.

Previously, a convergence theory has been provided that isaccurate concerning the dependence of the condition number on thegrid size. There, an auxiliary problem with p = 1 was introduced.The proof uses the fact that the stiffness matrices of theoriginal and the auxiliary problem are spectrally equivalent. Theconstants in this spectral equivalence are independent of the gridsize 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 splinespaces of interest and that does not use such an auxiliaryproblem. This allows us to get rid of the exponential dependencein p. We will discuss the dependence of all estimates on p.Numerical experiments, which con rm our theoretical findings, willbe presented.