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Seminar on Partial Differential Equations

usually takes place each Tuesday at 09:00 in IM, rear building, ground floor
Chair: Šárka Nečasová, Milan Pokorný

Adaptive solution strategy for Richards' equation based on Discontinuous Galerkin methods and mesh refinement

Jean-Baptiste Clément
Czech Technical University
Tuesday, 1. March 2022 - 9:00 to 10:00
Richards' equation describes flows in variably saturated porous media. Its solution is challenging since it is a parabolic equation with nonlinearities and degeneracies. In particular, many real-life problems are demanding because they can involve steep/heterogeneous hydraulics properties, dynamic  boundary conditions or moving sharp wetting fronts. In this regard, the aim is to design a robust and efficient numerical method to solve Richards’ equation. Towards this direction, the work presented here deals with Discontinuous Galerkin methods which are very flexible discretization schemes. They are combined with BDF methods to get high-order solutions. Built upon these desirable features, an adaptive mesh refinement strategy is proposed to improve Richards’ equation simulations. Examples such as the impoundment of a multi-material dam or the groundwater dynamics of sandy beaches illustrate the abilities of the... more

Sweeping process and its stability with applications to lattices of elasto-plastic springs

Ivan Gudoshnikov
Institute of Mathematics, Czech Academy of Sciences
Tuesday, 22. February 2022 - 9:00 to 10:00
Moreau's sweeping process is a class of non-smooth evolution problems invented to handle one-sided constraints in natural processes involving e.g. elastoplasticity, friction and thresholds in electicity and electomagnetism. The sweeping process can be viewed as a geometric generalization of hysteresis models. I will discuss its asymptotic properties, especially focusing on the case of a periodic input, as it leads to periodic outputs forming an attracting set.
Another focus will be the stress analysis of lattices of elasto-plastic springs via a finite-dimensional sweeping process (with illustrative examples). The mentioned asymptotic properties lead to nice conclusions about stress trajectories in the lattice models.
This is a joint project with Oleg Makarenkov, Dmitry Rachinskiy (University of Texas at Dallas) and Yang Jiao (Arizona State University).

Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable

Yong Lu
Nanjing University
Tuesday, 4. January 2022 - 9:00 to 10:00
We prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which is slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to  2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of constant state nor of small energy.

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