Speaker
Description
The mass matrix arising from the immersed isogeometric method is ill-conditioned due to the presence of severely trimmed elements in the computational domain, that is, elements of the mesh that only partially overlap with the domain of interest. We propose a preconditioning strategy based on the Additive Schwarz method.
Firstly, in contrast with existing methods, we use a preconditioner with Kronecker product structure rather than a simple diagonal scaling on the global space, to achieve a more robust preconditioner with respect to the polynomial degree. Secondly, we try a new approach for the construction of the Additive Schwarz local spaces, by considering basis functions with trimmed support, instead of trimmed elements. At the same time, since the local spaces associated to the trimmed degrees of freedom are more in number and larger than those related to trimmed elements, we investigate different strategies to select a subset of local spaces, with the aim of reducing the number of local spaces considered for the preconditioner, decreasing its computational cost, without affecting its performance.
Numerical experiments (in two dimensions) are considered to test the efficiency of the resulting preconditioners.
References:
[1] F. de Prenter, C. Verhoosel, G. van Zwieten, H. van Brummelen, ''Condition number analysis and preconditioning of the finite cell method'', Computer Methods in Applied Mechanics and Engineering 316 (2017).
[2] F. de Prenter, C. Verhoosel, H. van Brummelen, ''Preconditioning immersed isogeometric finite element methods with application to flow problems'', Computer Methods in Applied Mechanics and Engineering 348 (2019).
[3] G. Loli, G. Sangalli, M. Tani, ''Easy and efficient preconditioning of the isogeometric mass matrix'', Computers and Mathematics with Applications 116 (2022).
[4] F. de Prenter, C. Verhoosel, H. van Brummelen, M. Larson, S. Badia, ''Stability and conditioning of immersed finite element methods: analysis and remedies'', arXiv:2208.08538 (2022).