Mikael Mortensen


Numerical methods

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 [1] [2] [3].

We have also been studying the singular Neumann problem of linear elasticity [4]. 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.

References

  1. M. Kuchta, M. Nordaas, J. C. G. Verschaeve, M. Mortensen and K.-A. Mardal. Preconditioners for Saddle Point Systems With Trace Constraints Coupling 2D and 1D Domains, SIAM Journal on Scientific Computing, 38(6), pp. B962-B987, doi: 10.1137/15M1052822, 2016.
  2. M. Kuchta, K.-A. Mardal and M. Mortensen. Preconditioning Trace Coupled 3d-1d Systems Using Fractional Laplacian, Numer. Methods Partial Differential Equations, 35(1), pp. 375-393, doi: 10.1002/num.22304, 2018.
  3. M. Kuchta, K. A. Mardal and M. Mortensen. Characterization of the Space of Rigid Motions in Arbitrary Domains, 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), 2015.
  4. M. Kuchta, K.-A. Mardal and M. Mortensen. On the Singular Neumann Problem in Linear Elasticity, Numerical Linear Algebra with Applications, 26(1), pp. e2212, doi: 10.1002/nla.2212, 2018.