arXiv subjects

Jan Blechta

Publications and source records attributed to Jan Blechta.

5 recordsLinked to original sources

arXiv2025

On bifurcations and traction forces on an obstacle in incompressible flow

A systematic numerical investigation of flow-regime transitions in the two-dimensional incompressible Navier-Stokes flow past a confined circular cylinder is presented. For a fixed benchmark geometry, we observe a clear empirical correspondence between qualitative changes in steady traction profiles, understood here as the pointwise force density given by the Cauchy stress tensor on the obstacle boundary, and bifurcations in the long-time behavior of the unsteady Navier-Stokes equations. The observed transitions include onset of time-periodic oscillations, the appearance of multiple steady solutions and loss of effective symmetry. The well-known planar Sch\"afer-Turek benchmark is considered for Reynolds numbers up to 500. Several numerical techniques are employed to compute steady solutions, boundary traction profiles, and linear stability spectra such as duality-based approach for traction evaluation, deflation methods for detecting multiple steady states, and both two- and three- dimensional linear stability analyzes. The results suggest that steady boundary traction profiles can serve as a sensitive diagnostic indicator of critical Reynolds numbers at which qualitative changes in flow dynamics occur. This suggests a computationally inexpensive, complementary approach for detecting flow-regime transitions within this benchmark configuration.

physics.flu-dyn↗

arXiv2024

Fast construction of the discrete Green operator for a second order ordinary differential equation

We consider linear second order differential equation y''= f with zero Dirichlet boundary conditions. At the continuous level this problem is solvable using the Green function, and this technique has a counterpart on the discrete level. The discrete solution is represented via an application of a matrix -- the Green matrix -- to the discretised right-hand side, and we propose an algorithm for fast construction of the Green matrix. In particular, we discretise the original problem using the spectral collocation method based on the Chebyshev--Gauss--Lobatto points, and using the discrete cosine transformation we show that the corresponding Green matrix is fast to construct even for large number of collocation points/high polynomial degree. Furthermore, we show that the action of the discrete solution operator (Green matrix) to the corresponding right-hand side can be implemented in a matrix-free fashion.

math.NA↗

arXiv2022

Efficient solution of parameter identification problems with $H^1$ regularization

We consider the identification of spatially distributed parameters under $H^1$ regularization. Solving the associated minimization problem by Gauss-Newton iteration results in linearized problems to be solved in each step that can be cast as boundary value problems involving a low-rank modification of the Laplacian. Using algebraic multigrid as a fast Laplace solver, the Sherman-Morrison-Woodbury formula can be employed to construct a preconditioner for these linear problems which exhibits excellent scaling w.r.t. the relevant problem parameters. We first develop this approach in the functional setting, thus obtaining a consistent methodology for selecting boundary conditions that arise from the $H^1$ regularization. We then construct a method for solving the discrete linear systems based on combining any fast Poisson solver with the Woodbury formula. The efficacy of this method is then demonstrated with scaling experiments. These are carried out for a common nonlinear parameter identification problem arising in electrical resistivity tomography.

math.NA↗

arXiv2020

Stability of linear GMRES convergence with respect to compact perturbations

Suppose that a linear bounded operator $B$ on a Hilbert space exhibits at least linear GMRES convergence, i.e., there exists $M_B<1$ such that the GMRES residuals fulfill $\|r_k\|\leq M_B\|r_{k-1}\|$ for every initial residual $r_0$ and step $k\in\mathbb{N}$. We prove that GMRES with a compactly perturbed operator $A=B+C$ admits the bound $\|r_k\|/\|r_0\|\leq\prod_{j=1}^k\bigl(M_B+(1+M_B)\,\|A^{-1}\|\,\sigma_j(C)\bigr)$, i.e., the singular values $\sigma_j(C)$ control the departure from the bound for the unperturbed problem. This result can be seen as an extension of [I. Moret, A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal., 34 (1997), pp. 513-516, https://doi.org/10.1137/S0036142993259792], where only the case $B=\lambda I$ is considered. In this special case $M_B=0$ and the resulting convergence is superlinear.

math.NA↗

arXiv2019

On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion

In the first part of the paper we provide a new classification of incompressible fluids characterized by a continuous monotone relation between the velocity gradient and the Cauchy stress. The considered class includes Euler fluids, Navier-Stokes fluids, classical power-law fluids as well as stress power-law fluids, and their various generalizations including the fluids that we refer to as activated fluids, namely fluids that behave as an Euler fluid prior activation and behave as a viscous fluid once activation takes place. We also present a classification concerning boundary conditions that are viewed as the constitutive relations on the boundary. In the second part of the paper, we develop a robust mathematical theory for activated Euler fluids associated with different types of the boundary conditions ranging from no-slip to freeslip and include Navier's slip as well as stick-slip. Both steady and unsteady flows of such fluids in three-dimensional domains are analyzed.

math.AP↗