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