A preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis
For researchers in isogeometric analysis, this provides an efficient preconditioning strategy for nonsymmetric systems, though it is an incremental extension of an existing method.
This work extends a robust direct solver-based preconditioner, previously effective for symmetric Galerkin isogeometric systems, to nonsymmetric linear systems arising from collocation and weighted quadrature discretizations of the Poisson problem. Numerical experiments on 2D and 3D problems demonstrate the preconditioner's efficiency and robustness with respect to mesh size and spline degree.
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.