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

## Dr. Stefan Takacs

**Nov. 17, 2020, 2:30 p.m. ZOOM**

Isogeometric Analysis (IgA) is a spline-based finite element approach with global geometry function. Following the usual approach, the overall computational domain is 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 the solution of that linear system, domain decomposition approaches are a canonical choice since they can be based on the subdivision of the overall domain into patches. We consider the Isogeometric Analysis Tearing and Interconnecting (IETI) method,which is a variant of the FETI-method for IgA. 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. Numerical experiments, which confirm our theoretical findings, will be presented.