# A preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis

## TL;DR

This paper introduces a preconditioning strategy for solving nonsymmetric linear systems from isogeometric analysis discretizations, improving efficiency and robustness in 2D and 3D problems.

## Contribution

It extends a known tensor-structured solver preconditioning approach to nonsymmetric systems from collocation and weighted quadrature methods in isogeometric analysis.

## Key findings

- Preconditioner is effective for 2D and 3D problems.
- Preconditioner shows robustness with respect to mesh size and spline degree.
- Numerical experiments confirm improved solver efficiency.

## Abstract

In the context of isogeometric analysis, we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self-adjoint. These are the collocation method and the recently-developed weighted quadrature for the Galerkin discretization. In this paper, we are interested in the solution of the linear systems arising from the discretization of the Poisson problem using one of these approaches. In [SIAM J. Sci. Comput. 38(6) (2016) pp. A3644--A3671], a well-established direct solver for linear systems with tensor structure was used as a preconditioner in the context of Galerkin isogeometric analysis, yielding promising results. In particular, this preconditioner is robust with respect to the mesh size $h$ and the spline degree $p$. In the present work, we discuss how a similar approach can applied to the considered nonsymmetric linear systems. The efficiency of the proposed preconditioning strategy is assessed with numerical experiments on two-dimensional and three-dimensional problems.