Stability properties of the \(L^2\)-projection mapping to finite element spaces on adaptively generated meshes
Dienstag, 11.7.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\n\nThe \(L^2\)-projection mapping to Lagrange finite element spaces is a crucial tool in numerical analysis. Its Sobolev stability is known to be the key to discrete stability and quasi-optimality estimates for parabolic problems. For adaptively generated meshes the proof of Sobolev stability is challenging and requires conditions on how strongly the mesh size varies.\n\nHence, for the newest vertex bisection and its generalisation to higher dimensions by Maubach and Traxler we present optimal estimates on the mesh grading. Previously, grading estimates have been available only for 2D mesh refinement strategies. For such adaptively generated meshes we discuss Sobolev stability of the \(L^2\)-projection mapping to Lagrange finite element spaces under certain conditions on the polynomial degree and on the space dimension. In particular, the \(L^2\)-projection is \(W^{1,2}\)-stable for any polynomial degree, for any space dimension smaller than \(7\).\n\nThis is joint work with Lars Diening and Johannes Storn (Bielefeld University).\n
On the role of discrete Green’s operator preconditioning in FFT-based computational homogenization methods
Dienstag, 25.7.23, 15:00-16:00, Raum 226, Hermann-Herder-Str. 10
Solving computational homogenization problems on fine grids leads to systems of linear equations with millions to billions of unknowns, which favour iterative solvers over direct solvers.\nHowever, the number of iterations of iterative solvers can grow with increasing system size. To\novercome this issue, the so-called FFT-based solvers use the discrete Green’s operator preconditioning, which makes the condition number of the resulting linear system independent of the\nsystem/grid size [1, 2]. We studied the discrete Green’s operator preconditioning from a linear\nalgebra viewpoint and showed that all individual eigenvalues of such preconditioned systems can\nbe bounded purely from the knowledge of the material data of the problems, both original and\nreference. We developed a simple algorithm to compute these bounds [3, 4]. In my talk, I will\ndiscuss the theoretical aspects of these results and practical applications of the discrete Green’s\noperator preconditioning to periodic homogenisation problems discretised on regular grids [5].\n\n\nReferences\n\n[1] Moulinec, H. and Suquet, P., A fast numerical method for computing the linear and nonlinear\nmechanical properties of composites, Comptes Rendus de l’Acad´emie des sciences. S´erie II.\nM´ecanique, physique, chimie, astronomie, 318 (1994) 1417–1423\n\n[2] Schneider, M., A review of nonlinear FFT-based computational homogenization methods,\nActa Mechanica, 29 (2021) DOI 10.1007/s00707-021-02962-1,\n\n[3] Ladeck´y, M. and Pultarov´a, I. and Zeman, J., Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method,\nApplications of Mathematics, 66, 21–42 (2020) DOI 10.21136/AM.2020.0217-19\n\n[4] Pultarov´a, I., Ladeck´y, M., Two-sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems, Numerical Linear Algebra Applications\n28 (2021) e2382. DOI 10.1002/nla.2382.\n\n[5] Ladeck´y, M., Leute, J.R., Falsafi, A., Pultarov´a, I., Pastewka, L., Junge, T., and Zeman,\nJ. An Optimal Preconditioned FFT-accelerated Finite Element Solver for Homogenization.\nApplied Mathematics and Computation 446 (2023) 127835 DOI 10.1016/j.amc.2023.127835