Numerical methods#
In my recent paper [Mortensen, 2023] I describe a new, highly efficient and generic spectral Petrov-Galerkin method that always leads to strictly banded and well conditioned coefficient matrices. The method can be understood as a Petrov-Galerkin description of the integration preconditioner method by Coutsias et al. (1995).
My (now finished) PhD student Miroslav Kuchta has been looking at numerical methods to solve saddle point systems arising from trace constraints coupling 2D and 1D domains, or 3D and 1D domains [Kuchta et al., 2015, Kuchta et al., 2018, Kuchta et al., 2016].
We have also been studying the singular Neumann problem of linear elasticity [Kuchta et al., 2018]. Four different formulations of the problem have been analyzed and mesh independent preconditioners established for the resulting linear systems within the framework of operator preconditioning. We have proposed a preconditioner for the (singular) mixed formulation of linear elasticity, that is robust with respect to the material parameters. Using an orthonormal basis of the space of rigid motions, discrete projection operators have been derived and employed in a modification to the conjugate gradients method to ensure optimal error convergence of the solution.
With colleagues at the Extreme Computing Research Center (ECRC), King Abdullah University of Science and Technology (KAUST), we have been using spectralDNS to investigate time integration of Fourier pseudospectral Direct Numerical Simulations [Ketcheson et al., 2020]. We investigate the use of higher‐order Runge‐Kutta pairs and automatic step size control based on local error estimation. We find that the fifth‐order accurate Runge‐Kutta pair of Bogacki and Shampine gives much greater accuracy at a significantly reduced computational cost.
References#
David I. Ketcheson, Mikael Mortensen, Matteo Parsani, and Nathanael Schilling. More efficient time integration for fourier pseudospectral dns of incompressible turbulence. International Journal for Numerical Methods in Fluids, 92(2):79–93, 2020. doi:10.1002/fld.4773.
M. Kuchta, K. A. Mardal, and M. Mortensen. Characterization of the space of rigid motions in arbitrary domains. In In Bjørn Helge Skallerud and Helge Ingolf Andersson (ed.), MekIT'15 - Eight national conference on Computational Mechanics. International Center for Numerical Methods in Engineering (CIMNE), 259–274. 2015.
Miroslav Kuchta, Kent-Andre Mardal, and Mikael Mortensen. On the singular neumann problem in linear elasticity. Numerical Linear Algebra with Applications, 2018. URL: https://arxiv.org/abs/1609.09425.
Miroslav Kuchta, Kent-Andre Mardal, and Mikael Mortensen. Preconditioning trace coupled 3d-1d systems using fractional Laplacian. Numer. Methods Partial Differential Equations, 2018. URL: https://arxiv.org/abs/1612.03574.
Miroslav Kuchta, Magne Nordaas, Joris C. G. Verschaeve, Mikael Mortensen, and Kent-Andre Mardal. Preconditioners for saddle point systems with trace constraints coupling 2d and 1d domains. SIAM Journal on Scientific Computing, 38(6):B962–B987, 2016. doi:10.1137/15M1052822.
Mikael Mortensen. A generic and strictly banded spectral petrov–galerkin method for differential equations with polynomial coefficients. SIAM Journal on Scientific Computing, 45(1):A123–A146, 2023. doi:10.1137/22M1492842.