# A Discontinuous Galerkin Dual-Primal Isogeometric Tearing and Interconnecting Method with Inexact Local Solvers

## DI Rainer Schneckenleitner

**March 10, 2020, 3:30 p.m. S2 416-1**

Isogeometric discretizations of partial differential equations often lead to large-scale

linear systems. A class of methods that provide efficient solution strategies for those systems

are iterative substructuring methods. We propose a Dual-Primal Isogeometric Tearing

and Interconnecting (IETI-DP) method which was introduced in [1] and further analyzed,

e.g., in [2]. Usually, the local subproblems in IETI-DP are solved with sparse direct solvers.

If the local subproblems are very large or if a high polynomial degree is used single processors

could run out of memory. In such a case iterative solvers are superior to sparse

direct solvers regarding the memory requirement. To obtain fast solvers for Isogeometric

Analysis (IgA) the tensor product structure of the involved spaces should be exploited. For

continuous Galerkin discretizations the local problems have tensor product structure and

fast tensor product based preconditioning techniques can be used, cf. [3]. In case of discontinuous

Galerkin discretizations in IgA the local problems do not have tensor product

structure. To circumvent this issue, we propose a preconditioner that is based on an additive

Schwarz approach by splitting the local spaces into two subspaces. One subspace corresponds

to the inner degrees of freedom were the tensor product structure can be exploited

using the fast diagonalization method. In the second subspace we solve a small problem

which corresponds to an edge in 2D. We investigate the solvability of the local problems

with inexact solvers and we provide numerical examples.

References

[1] Kleiss, S., Pechstein, C., Jüttler, B., and Tomar S. IETI-Isogeometric Tearing and

Interconnecting Comput. Methods Appl. Mech. Eng. vol. 247–248 (2012), pp. 201–

215.

[2] Hofer, C., and Langer, U. Dual-Primal Isogeometric Tearing and Interconnecting

Methods. In Contributions to Partial Differential Equations and Applications, ed. by

B. N. Chetverushkin et al., Springer-ECCOMAS series “Computational Methods in

Applied Sciences”, vol. 47 (2019), pp. 273–296.

[3] Hofer, C., Langer, U. and Takacs, S. Inexact Dual-Primal Isogeometric Tearing and

Interconnecting Methods. In: Domain Decomposition Methods in Science and Engineering

XXIV, ed by Bjørstad P. et al., DD 2017. Lecture Notes in Computational

Science and Engineering, vol. 125 (2018), pp. 393–403, Springer, Cham